hcju/isabelleps huggingface dataset:
sym_iff_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x \<in> P"   and "y \<in> P" shows "(x \<sim> y) = (x = y \<and> y \<sqsubseteq> y)"
not_ev: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> ev \<phi> xs) = alw (\<lambda>xs. \<not> \<phi> xs)"
simps_5: fixes t :: "MIR.num" shows "not (Le t) = Gt t"
d_in: shows "D \<in> N_of_state (lnth Sts i)"
nxt_both_repeatable: fixes t' :: "'b ttree"   and x :: "'b"   and t :: "'b ttree" assumes "repeatable t'"   and "possible t' x" shows "nxt (t' \<otimes>\<otimes> t) x = t' \<otimes>\<otimes> (t \<oplus>\<oplus> nxt t x)"
assms_4: fixes K :: "'a set" assumes "subring K R"   and "H \<subseteq> K" shows "E \<subseteq> K"
simplex_def: fixes cs :: "constraint list" shows "simplex cs = simplex_index (zip [0..<length cs] cs)"
cost_swapFourSyms_le: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "consistent t"   and "minima t a b"   and "c \<in> alphabet t"   and "d \<in> alphabet t"   and "depth t c = height t"   and "depth t d = height t"   and "c \<noteq> d" shows "cost (swapFourSyms t a b c d) \<le> cost t"
simps_2: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and xs :: "'b list"   and x :: "'b" shows "sinks_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then insert (D x) (sinks_aux I D U xs) else sinks_aux I D U xs)"
ileI1: fixes m :: "enat"   and n :: "enat" assumes "m < n" shows "eSuc m \<le> n"
nonzero_sndI: fixes x :: "'b \<times> 'a" assumes "snd x \<noteq> (0::'a)" shows "x \<noteq> 0"
subset_imp: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and B :: "'e set" assumes "semilattice_order_set f less_eq less"   and "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "less_eq (semilattice_set.F f B) (semilattice_set.F f A)"
ls_nds_rel_def: fixes R :: "('a \<times> ('b, 'c) node_scheme) set" shows "\<langle>R\<rangle>ls_nds_rel = \<langle>R\<rangle>list_set_rel \<inter> {(uu_, s). inj_on name s}"
distinct_17: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "INTEG \<noteq> PRODUCT x51 x52"
env_map_env: fixes w :: "'a list" shows "env (map ENV w) = map ENV w"
big_eq_small: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: cs \<Rightarrow> t = small_steps \<pi> cs (Some (SKIP, t))"
Group_def: fixes G :: "('c, 'd) Group_scheme" shows "Group G \<equiv> ((\<cdot>\<^bsub>G\<^esub>) \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G \<and> (\<forall>a b c. a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> c \<in> carrier G \<longrightarrow> a \<cdot>\<^bsub>G\<^esub> b \<cdot>\<^bsub>G\<^esub> c = a \<cdot>\<^bsub>G\<^esub> (b \<cdot>\<^bsub>G\<^esub> c)) \<and> iop G \<in> carrier G \<rightarrow> carrier G) \<and> (\<forall>a. a \<in> carrier G \<longrightarrow> \<rho>\<^bsub>G\<^esub> a \<cdot>\<^bsub>G\<^esub> a = \<one>\<^bsub>G\<^esub>) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>a. a \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<cdot>\<^bsub>G\<^esub> a = a)"
inv_mult_group: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> y) = inv\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x"
simps_2: fixes m :: "'c \<Rightarrow> 'd set"   and ms :: "('c \<Rightarrow> 'd set) list" shows "\<Union>.m # ms = m \<union>. \<Union>.ms"
simps_2: fixes f :: "ZF"   and e :: "(ZF, ZF) Expression" shows "Expr2ZF f E@ e = ZFTriple (SucNat (ZFTFst (Expr2ZF e))) (nat2Nat 1) (Opair f (Expr2ZF e))"
less_option_None: fixes x :: "'b option" shows "(x < None) = False"
finite_cball_avoid: fixes S :: "'a set"   and X :: "'a set"   and p :: "'a" assumes "open S"   and "finite X"   and "p \<in> S" shows "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)"
Parallel_empty_lemma: fixes s :: "'b"   and Ts :: "('b ann_com option \<times> 'b set) list"   and t :: "'b"   and n :: "nat" assumes "(Parallel [], s) -Pn\<rightarrow> (Parallel Ts, t)" shows "Ts = [] \<and> n = 0 \<and> s = t"
simps_1: fixes f :: "'b \<Rightarrow> 'a option" shows "Option.bind None f = None"
3: shows "\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])) (dual\<^sub>s\<^sub>t S)"
radical_mult_distrib: fixes k :: "nat"   and r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a fps"   and b :: "'a fps" assumes "0 < k"   and "r k (fps_nth a 0) ^ k = fps_nth a 0"   and "r k (fps_nth b 0) ^ k = fps_nth b 0"   and "fps_nth a 0 \<noteq> (0::'a)"   and "fps_nth b 0 \<noteq> (0::'a)" shows "(r k (fps_nth (a * b) 0) = r k (fps_nth a 0) * r k (fps_nth b 0)) = (fps_radical r k (a * b) = fps_radical r k a * fps_radical r k b)"
many1_def: fixes tag :: "char list"   and p1 :: "xml \<Rightarrow> char list +\<^sub>\<bottom> 'f"   and p2 :: "xml \<Rightarrow> char list +\<^sub>\<bottom> 'g" shows "Xmlt.many1 tag p1 p2 = many1_gen tag p1 (\<lambda>_. p2)"
is_glb_unique: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "S >>| x"   and "S >>| y" shows "x = y"
Imp_I: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A H \<turnstile> B" shows "H \<turnstile> A IMP B"
discr_MtransT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>*t s'" shows "s \<approx> s'"
octo_of_real_inverse_collapse_1: fixes c :: "real" assumes "c \<noteq> 0" shows "octo_of_real c * octo_of_real (inverse c) = 1"
assms_3: shows "InvariantEquivalentZL (getF state) (getM state) F0"
replace_def: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) mapping" shows "replace k v m = (if k \<in> keys m then update k v m else m)"
j: shows "is_zero_row_upt_k j (mod_type_class.to_nat k) A"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "\<tau>multithreaded_wf final1 r1 \<tau>move1"   and "\<tau>multithreaded_wf final2 r2 \<tau>move2" shows "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2"
x'_def: shows "x' = (if nat \<bar>x\<bar> mod n \<le> (n - 1) div 2 then nat \<bar>x\<bar> mod n else n - nat \<bar>x\<bar> mod n)"
eq_fold: fixes g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "sum g A = Finite_Set.fold ((+) \<circ> g) (0::'d) A"
that: fixes e :: "real"   and B :: "real" assumes "0 < e"   and "e < 1"   and "0 < B"   and "(\<forall>z\<in>ball 0 e - {0}. cmod (f z) \<le> B) \<or> (\<forall>z\<in>ball 0 e - {0}. B \<le> cmod (f z))" shows "thesis"
npI': fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "\<exists>z. u \<cdot> z = v" shows "u \<le>np v"
minimum_segment_of_sub: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set"   and d :: "'c"   and m :: "'c" assumes "Torder D"   and "X \<subseteq> carrier D"   and "minimum_elem D (segment (Iod D X) d) m" shows "minimum_elem D X m"
gauss_int_code_post_10: fixes b :: "num" shows "Gauss_Int (numeral b) 0 = numeral b"
redundant_infer_def: fixes Inf :: "'a inference set"   and Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N :: "'a set"   and \<iota> :: "'a inference" assumes "calculus_with_standard_redundancy Inf Bot entails" shows "calculus_with_standard_redundancy.redundant_infer entails N \<iota> = (\<exists>DD\<subseteq>N. entails (DD \<union> set (side_prems_of \<iota>)) {concl_of \<iota>} \<and> (\<forall>D\<in>DD. D < main_prem_of \<iota>))"
div_free_divergence_refine_2: fixes P :: "'b process" shows "(D P = {}) = CHAOS UNIV \<sqsubseteq>\<^sub>D P"
cong_sym: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "[b = c] (mod a)" shows "[c = b] (mod a)"
explicit_info_set_def: fixes X0 :: "'h set" shows "explicit_info_set X0 = isEmpty_spec X0 \<bind> (\<lambda>e. WHILE\<^bsup>\<lambda>(e, X, Y). (e \<longrightarrow> X = {}) \<and> X0 = X \<union> (\<Union>(sctn, IS)\<in>Y. IS)\<^esup> (\<lambda>(e, X, Y). \<not> e) (\<lambda>(e, X, Y). split_with_info X \<bind> (\<lambda>(sctn, S, X'). isEmpty_spec X' \<bind> (\<lambda>e. RETURN (e, X', insert (sctn, S) Y)))) (e, X0, {}) \<bind> (\<lambda>(uu_, uu_, Xis). RETURN Xis))"
prime_factorsI: 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"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p x" shows "p \<in># factorial_semiring.prime_factorization zero times one normalize x"
that: fixes z :: "complex" assumes "z \<in> inside (path_image (p +++ linepath (a - complex_of_real d) (a + complex_of_real e)))"   and "cmod (winding_number (p +++ linepath (a - complex_of_real d) (a + complex_of_real e)) z) = 1" shows "thesis"
finite_out_arcs: fixes v :: "'a" shows "finite (out_arcs G v)"
integral_open_interval: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "integral (box a b) f = integral (cbox a b) f"
neq_Nil_conv: fixes xs :: "'b list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = Cons y ys)"
Inf_eqI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x" shows "\<Sqinter> A = x"
capacity_const: shows "\<forall>e. (0::'a) \<le> f e \<and> f e \<le> c e"
p_lt_1: fixes p :: "real"   and M :: "bool stream measure" assumes "infinite_coin_toss_space p M" shows "p \<le> 1"
os_is_it_def: fixes l :: "'a list"   and p :: "'a node ref option"   and l2 :: "'a list"   and it :: "'a node ref option" shows "os_is_it l p l2 it \<equiv> \<exists>\<^sub>Al1. \<up> (l = l1 @ l2) * lseg l1 p it * os_list l2 it"
norm_add_leD: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (plus a b) \<le> c" shows "norm b \<le> norm a + c"
eq_numeral_Suc: fixes k :: "num"   and n :: "nat" shows "(numeral k = Suc n) = (pred_numeral k = n)"
simps_1: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "pInput M xvec N\<cdot>P = M\<lparr>\<lambda>*xvec N\<rparr>.P"
Rep_inverse: fixes x :: "'b word" shows "word_of_int (sint x) = x"
select_convs_1: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
double_complement: fixes A :: "'d set" shows "- (- A) = A"
a_rcosI: fixes G :: "('e, 'f) ring_scheme"   and h :: "'e"   and H :: "'e set"   and x :: "'e" assumes "abelian_group G"   and "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<oplus>\<^bsub>G\<^esub> x \<in> H +>\<^bsub>G\<^esub> x"
is_singleton_def: fixes A :: "'a set" shows "is_singleton A = (\<exists>x. A = {x})"
dvd_mult_unit_iff': fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "(a dvd b * c) = (a dvd c)"
bounded_by_mono_alt: fixes b :: "real"   and Q :: "'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b Q"   and "P \<tturnstile> Q" shows "bounded_by b P"
pwd_owner_ii_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_owner_ii n of (s, S, A, B, C, D) \<Rightarrow> pwd_owner_ii_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
shiftr_def: fixes w :: "'b word"   and n :: "nat" shows "w >> n = (shiftr1 ^^ n) w"
allow_all_def: fixes p :: "'b" shows "A\<^sub>U p = \<lfloor>allow ()\<rfloor>"
fmeasurable_Diff: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> fmeasurable M"   and "B \<in> sets M" shows "A - B \<in> fmeasurable M"
weak_refl: fixes x :: "'a"   and y :: "'a" assumes "x .=\<^bsub>L\<^esub> y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> y"
higher_lower_decomp: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "higher p v + monomial (lookup p v) v + lower p v = p"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = (\<not> ((\<exists>nat. x = enat nat \<and> \<not> P (f1 nat)) \<or> x = \<infinity> \<and> \<not> P f2))"
fref: fixes K :: "('b \<times> 'd) set"   and V :: "('e \<times> 'f) set" assumes "single_valued K"   and "single_valued (K\<inverse>)" shows "(uncurry2 (RETURN \<circ>\<circ>\<circ> op_map_update), uncurry2 (RETURN \<circ>\<circ>\<circ> op_map_update)) \<in> (K \<times>\<^sub>r V) \<times>\<^sub>r \<langle>K, V\<rangle>map_rel \<rightarrow>\<^sub>f \<langle>\<langle>K, V\<rangle>map_rel\<rangle>nres_rel"
discrete_topology_closure_of: fixes U :: "'c set"   and S :: "'c set" shows "discrete_topology U closure_of S = U \<inter> S"
HPhase1or2Write_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HInv1 s"   and "HPhase1or2Write s s' p d" shows "HInv1 s'"
dM0r: shows "M0 * \<rho> * adjoint M0 \<in> carrier_mat d d"
naturality: fixes f :: "'c" assumes "C.arr f" shows "\<epsilon>.map (C.cod f) \<cdot>\<^sub>C GF.map f = C.map f \<cdot>\<^sub>C \<epsilon>.map (C.dom f)"
assigns_comp: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" shows "\<langle>f\<rangle>\<^sub>a ;; \<langle>g\<rangle>\<^sub>a = \<langle>g \<circ> f\<rangle>\<^sub>a"
maximal_atlas': fixes c :: "('a, 'b) chart" assumes "\<And>c'. c' \<in> charts \<Longrightarrow> l-smooth_compat c c'"   and "domain c \<subseteq> carrier" shows "c \<in> l.atlas"
d_implies_ruc: fixes P :: "'c process" assumes "deterministic P" shows "ref_union_closed P"
chines_eq_1: shows "H\<mu>HH\<nu>\<pi>\<rho>.chine = \<mu>.leg0 \<down>\<down> HH\<nu>\<pi>\<rho>.leg1"
block_sat: fixes block :: "('d, 'e) fm list \<times> 'e"   and M :: "('f, 'd) model"   and g :: "'e \<Rightarrow> 'f"   and w :: "'f" assumes "\<forall>p. p on block \<longrightarrow> M, g, w \<Turnstile> p" shows "M, g \<Turnstile>\<^sub>B block"
ord_inj': fixes a :: "'a" assumes "a \<in> carrier (mult_of R)" shows "inj_on (([^]) a) {1..G.ord a}"
lfp_Kleene_iter: fixes f :: "'b \<Rightarrow> 'b"   and k :: "nat" assumes "mono f"   and "(f ^^ Suc k) bot = (f ^^ k) bot" shows "lfp f = (f ^^ k) bot"
non_empty_distribution: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and nr1 :: "nat"   and n1 :: "nat"   and A1 :: "'b list list"   and nc1 :: "nat"   and A2 :: "'b list list"   and nr2 :: "nat"   and n2 :: "nat"   and B1 :: "'b list list"   and nc2 :: "nat"   and B2 :: "'b list list" assumes "plus_mult id f zer g inver"   and "mat nr1 n1 A1"   and "mat n1 nc1 A2"   and "mat nr2 n2 B1"   and "mat n2 nc2 B2"   and "A1 \<noteq> []"   and "B1 \<noteq> []"   and "A2 \<noteq> []"   and "B2 \<noteq> []" shows "mult.Tensor f (mat_multI zer g f (mult.row_length A1) A1 A2) (mat_multI zer g f (mult.row_length B1) B1 B2) \<noteq> []"
scale_minus_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b" assumes "module scale" shows "scale (- a) x = - scale a x"
extRulesp_extRules_eq: fixes R :: "('a sequent list \<times> 'a sequent) set" shows "extRulesp (\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> R*)"
assms_2: shows "interpret_floatarith fa vs \<noteq> 0"
pnormalize_Cons: fixes h :: "'a"   and p :: "'a list" shows "pnormalize (h # p) = (if pnormalize p = [] then if h = (0::'a) then [] else [h] else h # pnormalize p)"
CompFunCarAudioSystem_the: shows "the (CompFun CarAudioSystem ''On'') = {CDMode_CTRL, TunerMode_CTRL}"
trans: fixes a :: "'e list"   and b :: "'e list"   and c :: "'e list" assumes "prefix a b"   and "prefix b c" shows "prefix a c"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "Left_a.seq g f" shows "Left_a.comp g f = g \<cdot> f"
ldrop_inf: fixes xs :: "'a llist" shows "ldrop \<infinity> xs = LNil"
the_inv_into_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "the_inv_into A f = (\<lambda>x. THE y. y \<in> A \<and> f y = x)"
fin_append_com_Nil2: fixes x :: "'d list"   and z :: "nat \<Rightarrow> 'd" shows "fin_inf_append x (fin_inf_append [] z) = fin_inf_append (x @ []) z"
ordertype_le_Ord: fixes \<alpha> :: "V"   and X :: "V set" assumes "Ord \<alpha>"   and "X \<subseteq> elts \<alpha>" shows "tp X \<le> \<alpha>"
fold_simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a" shows "RBT_Impl.fold f Empty = id"
MsgsCJ': shows "msgsCJ' = msgs cJ -# <pJ, inM False>"
summable_comparison_test': fixes g :: "nat \<Rightarrow> real"   and N :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "summable g"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n" shows "summable f"
finite_sup_insertI: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>l. is_lub L l (insert x A) \<Longrightarrow> P l"   and "finite A"   and "x \<in> carrier L"   and "A \<subseteq> carrier L" shows "P (\<Squnion>insert x A)"
oalist_inv_tl_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs" shows "oalist_inv (tl_raw xs)"
a_3_var: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y \<cdot> (x + y) = (0::'a)"
apex_def: shows "\<mu>\<nu>\<pi>.apex \<equiv> C.dom \<mu>\<nu>\<pi>.leg0"
Abs_uint_inverse: fixes y :: "dflt_size word" assumes "y \<in> UNIV" shows "Rep_uint (Abs_uint y) = y"
simps: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com"   and a2 :: "'b" shows "PL.mayDiverge tval aval a1 a2 = (\<exists>c s c' s'. a1 = c \<and> a2 = s \<and> PL.transC tval aval (c, s) (c', s') \<and> PL.mayDiverge tval aval c' s')"
ccNeighbors_ccSquare: fixes x :: "Vars.var"   and S :: "Vars.var set" shows "ccNeighbors x (S\<^sup>2) = (if x \<in> S then S else {})"
h0a: shows "dim_col (take_cols_from_matrix (alt_matrix_A_R signs subsets) S) = length (take_indices signs S)"
integrable_const_bound: fixes f :: "'a \<Rightarrow> 'b"   and B :: "real" assumes "AE x in M. norm (f x) \<le> B"   and "random_variable borel f" shows "integrable M f"
intros_16: fixes c1 :: "i64"   and c2 :: "i64"   and iop :: "relop_i" shows "\<lparr>[$C ConstInt64 c1, $C ConstInt64 c2, $Relop_i T_i64 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_relop_i iop c1 c2))]\<rparr>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) Cones.map \<Phi>.map"
relcompp_apply: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'f"   and c :: "'h" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
map_poly_pcompose: fixes f :: "int poly"   and g :: "int poly" shows "of_int_poly (f \<circ>\<^sub>p g) = of_int_poly f \<circ>\<^sub>p of_int_poly g"
set_access_gen_axioms: shows "set_access_gen set_of memb empty un gen_succ"
rtrancl_len_E: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set" assumes "(x, y) \<in> r\<^sup>*" obtains n :: "nat" where "(x, y) \<in> r ^^ n"
PivotUpdateMinVars_axioms: shows "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update_code"
Lambert_W'_def: fixes x :: "real" shows "Lambert_W' x = (if x \<in> {- exp (- 1)..<0} then THE w. w \<le> - 1 \<and> w * exp w = x else - 1)"
winding_number_lt_half_linepath: fixes z :: "complex"   and a :: "complex"   and b :: "complex" assumes "z \<notin> closed_segment a b" shows "\<bar>Re (winding_number (linepath a b) z)\<bar> < 1 / 2"
bounded: shows "\<exists>K. \<forall>xa. norm ((if x \<in> T then g' x else (\<lambda>x. 0::'b)) xa) \<le> norm xa * K"
tap_def: fixes f :: "heap \<Rightarrow> 'a" shows "tap f = Heap (\<lambda>h. Some (f h, h))"
abs_pat_alpha': fixes p :: "pat"   and t :: "btrm"   and q :: "pat"   and u :: "btrm" assumes "(\<lambda>[p]. t) = (\<lambda>[q]. u)"   and "pat_type p = pat_type q"   and "distinct (pat_vars p)"   and "distinct (pat_vars q)" shows "\<exists>pi. p = pi \<bullet> q \<and> t = pi \<bullet> u \<and> set pi \<subseteq> (supp p \<union> supp q) \<times> (supp p \<union> supp q)"
normalize_polynomial: fixes K :: "'a set"   and p :: "'a list" assumes "polynomial K p" shows "R.normalize p = p"
Un_Diff_Int: fixes A :: "'c set"   and B :: "'c set" shows "A - B \<union> A \<inter> B = A"
bv_to_nat0: fixes bs :: "bit list" shows "bv_to_nat (\<zero> # bs) = bv_to_nat bs"
assms_2: shows "f maps\<^bsub>A\<^esub> X to Y"
simps_7: fixes lformula0 :: "'a \<Rightarrow> bool"   and k :: "'b"   and \<phi> :: "('a, 'b) aformula" shows "Formula_Operations.lformula lformula0 (FAll k \<phi>) = Formula_Operations.lformula lformula0 \<phi>"
set_supp_eq: fixes xs :: "W.tvar list" shows "set xs = supp xs"
at_least_two_elements_def: fixes some_non_bottom_element :: "'a" shows "at_least_two_elements some_non_bottom_element \<equiv> some_non_bottom_element \<noteq> \<bottom>"
keys_subset_singleton_imp_monomial: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "keys p \<subseteq> {t}" shows "monomial (lookup p t) t = p"
ifex_ite_opt_eq: fixes i :: "'b ifex"   and t :: "'b ifex"   and e :: "'b ifex" assumes "ro_ifex i"   and "ro_ifex t"   and "ro_ifex e" shows "ifex_ite_opt i t e = ifex_ite i t e"
step: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto><(X \<union> weakBisim)> Q \<and> (Q, P) \<in> X \<union> weakBisim"
modular_2_aux': fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> - (z ; x\<^sup>T)) ; x \<cdot> z = 0"
path2_prefix: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and ns' :: "'node list"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "prefix (ns' @ [m']) ns" shows "g \<turnstile> n-ns' @ [m']\<rightarrow>m'"
intro: fixes \<U> :: "'b set"   and \<V> :: "'b set"   and \<A> :: "'b set multiset"   and \<B> :: "'b set multiset" assumes "\<U> \<subset> \<V>"   and "\<A> \<subseteq># \<B>" shows "proper_sub_set_system_axioms \<U> \<A> \<V> \<B>"
bdd_all_bdd_binop: fixes P :: "'a \<Rightarrow> bool"   and bdd :: "'a bdd"   and Q :: "'b \<Rightarrow> bool"   and bdd' :: "'b bdd"   and R :: "'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bdd_all P bdd"   and "bdd_all Q bdd'"   and "\<And>a b. \<lbrakk>P a; Q b\<rbrakk> \<Longrightarrow> R (f a b)" shows "bdd_all R (bdd_binop f bdd bdd')"
conforms_allocL: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and v :: "val"   and T :: "ty"   and vn :: "lname" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "G,s\<turnstile>v\<Colon>\<preceq>T" shows "(x, lupd(vn\<mapsto>v) s)\<Colon>\<preceq>(G, L(vn \<mapsto> T))"
removeAll_append: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"