hcju/isabelleps huggingface dataset:
power_minus_Bit1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus x) (numeral (num.Bit1 k)) = uminus (power.power one times x (numeral (num.Bit1 k)))"
sub_HdE: fixes s :: "('a, 'b) tm"   and \<xi> :: "('a, 'b) hd" assumes "sub s (Hd \<xi>)" obtains "Hd \<xi> = s"
Cd_Above_dX: fixes X :: "'a set"   and d :: "'b" assumes "PdXXX.dX X d \<subseteq> Field (Pd d)" shows "Cd d X = Above (Pd d) (PdXXX.dX X d) \<inter> X"
that: fixes Z :: "V" assumes "X \<approx>\<^sub>\<circ> Z"   and "Z \<inter>\<^sub>\<circ> Y = []\<^sub>\<circ>"   and "Z \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "thesis"
qFresh_qSwap_exchange2: fixes zs :: "'d"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and z2 :: "'e" shows "qFresh zs z1 X = qFresh zs z2 (X #[[z2 \<and> z1]]_zs)"
component_complement_connected: fixes S :: "'a set"   and C :: "'a set" assumes "connected S"   and "C \<in> components (- S)" shows "connected (- C)"
reflp_joinablep: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp P" shows "reflp (joinablep P)"
sel_rews_4: shows "appF\<cdot>ValFF = \<bottom>"
exists_bezout: fixes a :: "'a"   and b :: "'a" shows "\<exists>p q d. p * a + q * b = d \<and> d dvd a \<and> d dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd d)"
rbt_insertw_is_rbt: fixes t :: "('a, 'c) rbt"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c" assumes "is_rbt t" shows "is_rbt (rbt_insert_with f k v t)"
elims_3: fixes x :: "'b list \<times> 'b list" assumes "\<not> is_empty x" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "fs \<noteq> []"
Let_fresh: fixes a :: "atom"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "a \<sharp> Let \<Gamma> e = (a \<sharp> e \<and> a \<sharp> \<Gamma> \<or> a \<in> atom ` domA \<Gamma>)"
msed_map_invR: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and b :: "'b"   and N :: "'b multiset" assumes "image_mset f M = add_mset b N" shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> image_mset f M1 = N"
ex_gt_imp_less_multiset: fixes N :: "'b multiset"   and M :: "'b multiset" assumes "\<exists>y. y \<in># N \<and> (\<forall>x. x \<in># M \<longrightarrow> x < y)" shows "M < N"
TI': shows "set TI' = {(a, b). (a, b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop3 n (l, r) = (\<exists>i j k t. i + j = n \<and> 0 < i \<and> 0 < j \<and> k + t = Suc j \<and> (l, r) = (Bk \<up> k @ Oc \<up> i, Bk \<up> t @ Oc \<up> j))"
simps_3: fixes n :: "'d"   and t1 :: "'d ifex"   and t2 :: "'d ifex"   and s :: "'d \<Rightarrow> bool" shows "val_ifex (IF n t1 t2) s = (if s n then val_ifex t1 s else val_ifex t2 s)"
vars_mset_subst_mset_subset: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" shows "vars_mset (subst_mset (subst x t) M) \<subseteq> vars_mset M \<union> vars_term t \<union> {x}"
antecedent: shows "x \<oplus> y \<noteq> u"
inject: fixes x1 :: "oref \<Rightarrow> obj option"   and x2 :: "lname \<Rightarrow> val option"   and y1 :: "oref \<Rightarrow> obj option"   and y2 :: "lname \<Rightarrow> val option" shows "(st x1 x2 = st y1 y2) = (x1 = y1 \<and> x2 = y2)"
sum_le_suminf: fixes f :: "nat \<Rightarrow> 'c"   and I :: "nat set" assumes "summable f"   and "finite I"   and "\<And>n. n \<in> - I \<Longrightarrow> (0::'c) \<le> f n" shows "sum f I \<le> suminf f"
wp_def: fixes act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wp act B \<equiv> - act\<inverse> `` (- B)"
square_free_factorizationD_1: fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
lift_subst_bv2_dummy: fixes s :: "Core.term"   and i :: "nat" assumes "\<not> loose_bvar s i" shows "lift (decr i s) i = s"
semiring_norm_114: shows "(1::'a) \<le> (1::'a)"
ereal_zero_le_0_iff: fixes a :: "ereal"   and b :: "ereal" shows "(0 \<le> a * b) = (0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0)"
dim_eq_card: fixes B :: "'b set"   and V :: "'b set" assumes "span B = span V"   and "independent B" shows "dim V = card B"
multiplicative_function_def: fixes f :: "nat \<Rightarrow> 'a" shows "multiplicative_function f \<equiv> f 0 = (0::'a) \<and> f 1 = (1::'a) \<and> (\<forall>a b. 1 < a \<longrightarrow> 1 < b \<longrightarrow> coprime a b \<longrightarrow> f (a * b) = f a * f b)"
list2policyR_Start: fixes p :: "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"   and list :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<in> dom (Cp a)" shows "Cp (list2policyR (a # list)) p = Cp a p"
prems_2: shows "l2 = (a # l2a) @ [h] @ l2b"
pt_fresh_bij_ineq: fixes pi :: "('b \<times> 'b) list"   and a :: "'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
allUses_in_allVars: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> allUses g n"   and "n \<in> set (\<alpha>n g)" shows "v \<in> allVars g"
L3_5ii_lemma2: fixes c1 :: "'a com"   and c2 :: "'a com"   and s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a"   and n :: "nat" assumes "(Seq c1 c2, s) -Pn\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "\<exists>y m Rs. (c1, s) -P*\<rightarrow> (Parallel Rs, y) \<and> All_None Rs \<and> (c2, y) -Pm\<rightarrow> (Parallel Ts, t) \<and> m \<le> n"
pocklington_lemma: fixes n :: "nat"   and q :: "nat"   and r :: "nat"   and a :: "nat"   and p :: "nat" assumes "2 \<le> n"   and "n - 1 = q * r"   and "[a ^ (n - 1) = 1] (mod n)"   and "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n"   and "prime p"   and "p dvd n" shows "[p = 1] (mod q)"
cop_1: shows "comm_monoid_mult_class.coprime (lead_coeff f) p"
covector_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.covector f = (\<forall>i j k. f (i, j) = f (k, j))"
sH_while_inv: fixes P :: "'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and X :: "('b \<times> 'b) set" assumes "\<forall>s. P s \<longrightarrow> I s"   and "\<forall>s. I s \<and> \<not> R s \<longrightarrow> Q s"   and "PRE I \<sqinter> R X POST I" shows "PRE P (WHILE R INV I DO X OD) POST Q"
trimonoD_mono: fixes B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "trimono B" shows "mono B"
hm_update_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable"   and k :: "'a"   and v :: "'b" shows "<is_hashmap m ht> hm_update k v ht <is_hashmap (m(k \<mapsto> v))>\<^sub>t"
Ssegment_le_mono: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (Ssegment D a \<subseteq> Ssegment D b)"
snth_siterate: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" shows "siterate f x !! n = (f ^^ n) x"
result: shows "local.dist_degree_factorize_main v w d res = facts"
add_right_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
coeffs_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "coeffs (map_poly f p) = strip_while ((=) (0::'a)) (map f (coeffs p))"
ta_seq_consist_simps_1: fixes P :: "'a prog"   and vs :: "'b \<times> addr_loc \<Rightarrow> ('b val \<times> bool) option" shows "ta_seq_consist P vs LNil = True"
exp_strong_induct: fixes P :: "'a \<Rightarrow> exp \<Rightarrow> bool"   and c :: "'a"   and exp :: "exp" assumes "\<And>var c. P c (Var var)"   and "\<And>exp var c. (\<And>c. P c exp) \<Longrightarrow> P c (App exp var)"   and "\<And>\<Gamma> exp c. \<lbrakk>atom ` domA \<Gamma> \<sharp>* c; \<And>c x. x \<in> domA \<Gamma> \<Longrightarrow> P c (the (map_of \<Gamma> x)); \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Terms.Let \<Gamma> exp)"   and "\<And>var exp c. \<lbrakk>{atom var} \<sharp>* c; \<And>c. P c exp\<rbrakk> \<Longrightarrow> P c (Lam [var]. exp)"   and "\<And>b c. P c (Bool b)"   and "\<And>scrut e1 e2 c. \<lbrakk>\<And>c. P c scrut; \<And>c. P c e1; \<And>c. P c e2\<rbrakk> \<Longrightarrow> P c (scrut ? e1 : e2)" shows "P c exp"
ls_def: shows "ls = snd (subset_dfs A (nfa_startnode A))"
simps: fixes vs :: "nat list"   and f :: "facetype" shows "final (Face vs f) = (case f of Final \<Rightarrow> True | Nonfinal \<Rightarrow> False)"
assms_3: shows "\<forall>\<^sub>F n in G. g n \<in> A"
prems_1: shows "list_all (coll 0 x) (y # ys)"
i_in_inter: shows "\<iota> \<in> (\<Inter>q\<in>Q. Red_I_\<G>_L_q q NL)"
RG_mult_closed: fixes x :: "('d, 'a) aezfun"   and y :: "('d, 'a) aezfun" assumes "x \<in> ActingGroup.group_ring"   and "y \<in> ActingGroup.group_ring" shows "x * y \<in> ActingGroup.group_ring"
obj_simps_1: fixes a :: "'a" assumes "C.obj a" shows "C.arr a"
bind_fmap: fixes f :: "'g \<rightarrow> 'f"   and xs :: "'g\<cdot>'e"   and k :: "'f \<rightarrow> 'd\<cdot>'e" shows "fmap\<cdot>f\<cdot>xs \<bind> k = xs \<bind> (\<Lambda> x. k\<cdot>(f\<cdot>x))"
cnf2_lemma: fixes x :: "ordinal" assumes "0 < x" shows "x mod \<omega> ** oLog \<omega> x < x"
T_insert_rw: shows "T ` insert x A = insert (T x) (T ` A)"
hyps_1: shows "\<Gamma>\<turnstile>c \<down> s"
funpow_mod_eq: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and m :: "nat" assumes "(f ^^ n) x = x"   and "0 < n" shows "(f ^^ (m mod n)) x = (f ^^ m) x"
INT_E: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set"   and a :: "'d" assumes "b \<in> \<Inter> (B ` A)" obtains "b \<in> B a"   | "a \<notin> A"
list_def: fixes h :: "'c \<Rightarrow> 'c"   and p :: "'c" shows "list h p = (SOME as. List h p as)"
timpl_closure'_timpls_rtrancl_eq: fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' (c\<^sup>*) = timpl_closure' c"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = \<phi>\<psi>.\<epsilon> x \<cdot>\<^sub>C F g"
stable_on_range': fixes d :: "'b"   and ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "Singleton_for_d.stable_on d ds X"   and "x \<in> X" shows "x \<in> Field (Pd_singletons_for_ds X {d} (Xd x))"
LeftDerivationLadder_L_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<beta> :: "symbol list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.LeftDerivationLadder \<NN> \<TT> \<RR> \<alpha> D L \<beta>"   and "length L = 1" shows "\<exists>i. LocalLexing.LeftDerivationFix \<NN> \<TT> \<RR> \<alpha> i D (ladder_last_j L) \<beta>"
degree_newton_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list" shows "degree (newton_interpolation_poly xs_ys) \<le> length xs_ys - 1"
indoc_set: fixes a :: "'item"   and u :: "'item list" assumes "indoc a u" shows "a \<in> set u"
edge_cap_positive: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> E" shows "(0::'capacity) < c (u, v)"
real_of_nat_float: fixes x :: "nat" shows "real x \<in> float"
comm_semiring_1_ow_def: fixes U :: "'b set"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "comm_semiring_1_ow U times one plus zero \<equiv> zero_neq_one_ow U one zero \<and> comm_semiring_0_ow U plus zero times \<and> comm_monoid_mult_ow U times one"
bit_mask_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) one) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> n < m)"
init: shows "\<forall>x xa. xa \<noteq> x \<longrightarrow> (\<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts xa = ext v \<and> restrict v (res ts) xa = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True)) \<and> True))"
mdecls_visible: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list" assumes "wf_prog wf_md P"   and "is_class P C"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" shows "\<exists>Mm. P \<turnstile> C sees_methods Mm \<and> (\<forall>(M, Ts, T, m)\<in>set ms. Mm M = \<lfloor>((Ts, T, m), C)\<rfloor>)"
dioph_ins_def: fixes m :: "int" shows "dioph_ins m = (\<lambda>(is, js). (is[int_encode m := Some (length js)], js @ [m]))"
inverse_functors_axioms: shows "inverse_functors (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G"
assms_10: shows "A = \<Union> (roots ` X (=))"
class_add_sup_loc: fixes P :: "jvm_prog"   and LT :: "ty err list"   and LT' :: "ty err list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> LT [\<le>\<^sub>\<top>] LT'"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> LT [\<le>\<^sub>\<top>] LT'"
cubic_root_radical_sqrt_steplemma: fixes P :: "real set"   and a :: "real"   and b :: "real"   and c :: "real"   and z :: "real"   and u :: "real"   and v :: "real"   and s :: "real" assumes "\<nat> \<subseteq> P"   and "\<forall>x\<in>P. - x \<in> P"   and "\<forall>x\<in>P. x \<noteq> 0 \<longrightarrow> 1 / x \<in> P"   and "\<forall>x\<in>P. \<forall>y\<in>P. x + y \<in> P"   and "\<forall>x\<in>P. \<forall>y\<in>P. x * y \<in> P"   and "a \<in> P"   and "b \<in> P"   and "c \<in> P"   and "z ^ 3 + a * z\<^sup>2 + b * z + c = 0"   and "u \<in> P"   and "v \<in> P"   and "s * s \<in> P"   and "z = u + v * s" shows "\<exists>w\<in>P. w ^ 3 + a * w\<^sup>2 + b * w + c = 0"
rep_eq: fixes x :: "double" shows "Code_Double.is_nan x = IEEE.is_nan (rep_double x)"
lebesgue_set_almost_fsigma: fixes S :: "'b set" assumes "S \<in> sets lebesgue" obtains C :: "'b set" and T :: "'b set" where "fsigma C" and "T \<in> null_sets lebesgue" and "C \<union> T = S" and "disjnt C T"
u'_delta: shows "u' \<delta> = ego2.s' 0"
m_closed: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P"
bags_exist: fixes v :: "'a" assumes "v \<in> V" shows "\<exists>t\<in>V\<^bsub>T\<^esub>. v \<in> bag t"
prems_2: shows "e \<in> set (elements (a # p'))"
union_or: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
assms: shows "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>r new_shadow_root_ptr"
proj_subset_if_no_label_1: fixes l :: "'d"   and A :: "('d strand_label \<times> 'e) list"   and l' :: "'d" assumes "\<not> list_ex (is_LabelN l) A" shows "set (proj l A) \<subseteq> set (proj l' A)"
opT: fixes U :: "'a set" assumes "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"
ImpLive_def: fixes p :: "PrIds" shows "ImpLive p = (WF(MClkFwd memCh crCh cst p)_c p \<and> SF(MClkReply memCh crCh cst p)_c p \<and> WF(RPCNext crCh rmCh rst p)_r p \<and> WF(RNext rmCh mm ires p)_m p \<and> WF(MemReturn rmCh ires p)_m p)"
Finite: shows "finite {{A \<union> B |A B. A \<subseteq> P \<and> A \<Turnstile>\<^sub>P \<phi> \<and> B \<subseteq> UNIV - P} |\<phi>. nested_propos \<phi> \<subseteq> P}"
derivativeMonotonic: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and A :: "(pi \<times> pi) set"   and B :: "(pi \<times> pi) set" assumes "derivative P Q a x A"   and "A \<subseteq> B" shows "derivative P Q a x B"
exp_two_pi_i': shows "exp (\<i> * (complex_of_real pi * 2)) = 1"
int_pow_pow: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "int"   and m :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
tendsto_divide: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "b \<noteq> (0::'c)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
non_inf_int_gt: shows "non_inf {(a, b). b < a}"
simps_25: shows "cospan_shape.comp cospan_shape.BT cospan_shape.AT = cospan_shape.Null"
Term_Eats: fixes x :: "hf"   and y :: "hf" assumes "Term x"   and "Term y" shows "Term (q_Eats x y)"
takeWhile_dropWhile_id: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "takeWhile P xs @ dropWhile P xs = xs"
word_Suc_leq: fixes k :: "'b word"   and x :: "'b word" assumes "k \<noteq> - 1" shows "(x < k + 1) = (x \<le> k)"
assms_1: shows "nofail (m0 \<bind> WHILE cond step)"
the_\<theta>_props_1: shows "\<guillemotleft>\<omega>.the_\<theta> : t\<^sub>0 \<star> \<omega>.chine \<Rightarrow> r\<^sub>0\<guillemotright>"
simps_2: fixes uv :: "'b"   and x :: "'a"   and \<Pi> :: "('b, 'a) chain" shows "comlist (\<lbrace> uv \<rbrace> \<cdot> x \<cdot> \<Pi>) = x # comlist \<Pi>"
map_idx_nth: fixes i :: "nat"   and xs :: "'e list"   and f :: "'e \<Rightarrow> nat \<Rightarrow> 'f"   and n :: "nat" assumes "i < length xs" shows "map_idx f xs n ! i = f (xs ! i) (n + i)"
abc: shows "a\<^sup>2 + b\<^sup>2 = c\<^sup>2"
distrib_left: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m"   and c :: "'m" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
d: shows "d = rat_of_int (lead_coeff (gcd F G))"
a: shows "\<not> is_final (steps0 (1, l, r) B n) \<and> steps0 (1, l, r) B (Suc n) = (0, l', r')"
assms_2: shows "\<forall>\<^sub>F x in F. f x \<in> S"
conf_f_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap" shows "conf_f P h \<equiv> \<lambda>(ST, LT) is (stk, loc, C, M, pc). P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is"
Cn_notnot: fixes \<phi> :: "'a" shows "Cn {.\<not> (.\<not> \<phi>)} = Cn {\<phi>}"
intro: assumes "\<And>x y. (E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y) \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> E (dom x) \<^bold>\<and> \<^bold>\<not> (\<^bold>\<not> (E (cod y)) \<^bold>\<or> dom x \<noteq> cod y))"   and "\<And>x. cod (dom x) \<cong> dom x"   and "\<And>y. dom cod y \<cong> cod y"   and "\<And>x. x \<cdot> (dom x) \<cong> x"   and "\<And>y. cod y \<cdot> y \<cong> y"   and "\<And>x y. dom x \<cdot> y \<cong> dom (dom x) \<cdot> y"   and "\<And>x y. cod x \<cdot> y \<cong> cod x \<cdot> cod y"   and "\<And>x y z. x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z" shows "AxiomsSet6"
rank_conjugate: fixes A :: "'b mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n d" shows "vec_space.rank n (conjugate A) = vec_space.rank n A"
aux: fixes n :: "nat"   and M :: "nat set" shows "(n \<in> Suc ` M) = (0 < n \<and> n - 1 \<in> M)"
assms_2: fixes k :: "nat" shows "stake k v \<preceq>\<^sub>F w (l k)"
ladder_cut_prefix_j: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L"   and "0 < length L" shows "ladder_j (ladder_cut_prefix d L) index = ladder_j L index - d"
elims_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "xa__ = []"   | x :: "'c" and ys :: "'c list" where "xa__ = x # ys" and "Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys"
of_real_linepath: fixes a :: "real"   and b :: "real"   and x :: "real" shows "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
bipartite_E: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma> x y" shows "x \<in> A \<Gamma> \<and> y \<in> B \<Gamma>"
vars2: shows "free_vars e2 \<subseteq> V \<union> V'"
insert_Key_image: fixes K :: "nat"   and KK :: "nat set"   and C :: "msg set" shows "insert (Key K) (Key ` KK \<union> C) = Key ` insert K KK \<union> C"
compl_ifte: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "- ifte a x y = ifte a (- x) (- y)"
rep_eq: fixes x :: "'a perm" shows "affected x = {a. x \<langle>$\<rangle> a \<noteq> a}"
extractFrameSimps_7: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "name"   and P :: "('f, 'd, 'e) psi" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.extractFrame SCompose SBottom (\<lparr>\<nu>x\<rparr>P) = \<lparr>\<nu>x\<rparr>assertionAux.extractFrame SCompose SBottom P"
sequence_to_failure_non_nil: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and xs :: "('d \<times> 'e) list" assumes "sequence_to_failure M1 M2 xs" shows "xs \<noteq> []"
simps: fixes p :: "'b pmf"   and a :: "'b cset" shows "cr_pmf_cset p a = (a = cset_pmf p)"
push_down_oldest_token_configuration: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat"   and x :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some x" shows "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"
ab_1_1: fixes k :: "nat" shows "1 \<le> aa k"
get_scdom_component_contains_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" obtains c :: "(_) object_ptr list" where "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c" and "set c \<subseteq> set sc"
exhaust: fixes y :: "'b node" obtains "y = SOURCE"   | x2 :: "'b" where "y = Inner x2"
valid_index_ind_in_set: fixes "is" :: "nat list"   and A :: "nat set"   and B :: "nat set" assumes "is \<lhd> nths dims A"   and "B \<subseteq> A" shows "nths is (ind_in_set A ` B) \<lhd> nths dims B"