hcju/isabelleps huggingface dataset:
matrix_vector_mult_matrix_matrix_mult_compose: fixes A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'd) vec" shows "(*v) (A ** B) = (*v) A \<circ> (*v) B"
mk_rtrancl_def: fixes init :: "'a list" shows "mk_rtrancl init \<equiv> mk_rtrancl_main init {}"
tautologyInTotalValuation: fixes clause :: "Literal list"   and valuation :: "Literal list" assumes "clauseTautology clause"   and "vars clause \<subseteq> vars valuation" shows "clauseTrue clause valuation"
assms_3: fixes I :: "'a set" assumes "I \<in> P" shows "I0 \<inter> I = {}"
TrueInSituation_def: fixes x :: "\<kappa>"   and p :: "\<o>" shows "x \<^bold>\<Turnstile> p \<equiv> Situation x \<^bold>& x \<^bold>\<Sigma> p"
times_closed: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a *\<^sub>o\<^sub>w b \<in> U"
semiring_norm_168: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "- numeral v + (- numeral w + y) = - numeral (v + w) + y"
numeral_power_int_eq_of_real_cancel_iff: fixes x :: "num"   and n :: "int"   and y :: "real" shows "(numeral x powi n = of_real y) = (numeral x powi n = y)"
cx0: shows "\<forall>i<length Vs. (0::'a) \<le> cx i"
rep_eq: shows "(*\<^sub>V) 1 = one_dim_iso"
closest_supchamber_def: fixes F :: "'a set"   and D :: "'a set" shows "closest_supchamber F D = (ARG_MIN (\<lambda>C. chamber_distance C D) C. chamber C \<and> order.greater_eq C F)"
rref_col_rank: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "col_rank A = card {column (LEAST n. A $ i $ n \<noteq> (0::'a)) A |i. row i A \<noteq> 0}"
distr_M_seq: fixes n :: "nat" shows "real_distribution (M_seq n)"
CESem_bot: fixes e :: "exp"   and \<sigma> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<sigma>\<^esub>)\<cdot>\<bottom> = \<bottom>"
Abs_FingerTree_inverse: fixes y :: "('a, 'b) FingerTreeStruc" assumes "y \<in> {t. FingerTreeStruc.ft_invar t}" shows "Rep_FingerTree (Abs_FingerTree y) = y"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_DF_refine_F: fixes A :: "'b set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F DF A"
frontier_interiors: fixes s :: "'b set" shows "frontier s = - interior s - interior (- s)"
counterAssignModelAssign: fixes gamma :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "counterAssign \<in> modelAssigns (counterModel gamma)"
index_unit_vec_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "unit_vec n i $ j = (if j = i then 1::'b else (0::'b))"
ereal_Lim_uminus: fixes f :: "'a \<Rightarrow> ereal"   and f0 :: "ereal"   and net :: "'a filter" shows "(f \<longlongrightarrow> f0) net = ((\<lambda>x. - f x) \<longlongrightarrow> - f0) net"
nf_unicity: fixes x :: "trm"   and a :: "trm"   and b :: "trm" assumes "x \<Down> a"   and "x \<Down> b" shows "a = b"
UnionI: fixes X :: "'d set"   and C :: "'d set set"   and A :: "'d" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
lfp_F2_o_F1_fp_cop_F: fixes ds :: "'b set" shows "lfp (F2 ds \<circ> F1) = fp_cop_F ds"
that: fixes ys' :: "'a list" assumes "ys = x # ys'"   and "strict_prefix xs ys'" shows "thesis"
cod_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A1.cod (A1.dom f) = A1.dom f"
obs_singleton: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and n :: "'b"   and S :: "'b set" assumes "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S) = {}"
isomorphic_reflexive: fixes f :: "'d" assumes "D.ide f" shows "D.isomorphic f f"
mu_update: fixes i' :: "nat"   and j' :: "nat" assumes "i' < m"   and "j' < m" shows "gs2.\<mu> i' j' = (if i' = i \<and> j' \<le> j then gs1.\<mu> i j' - rat_of_int c * gs1.\<mu> j j' else gs1.\<mu> i' j')"
preallocated_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "preallocated h \<equiv> \<forall>C\<in>sys_xcpts. \<exists>S. h (addr_of_sys_xcpt C) = \<lfloor>(C, S)\<rfloor>"
unique_NB: fixes A :: "agent"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list"   and A' :: "agent"   and NA' :: "nat" assumes "Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (pubK A') \<lbrace>Nonce NA', Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "Nonce NB \<notin> analz (knows Spy evs)"   and "evs \<in> ns_public" shows "A = A' \<and> NA = NA'"
induct: fixes P :: "('f \<times> 'd) list \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> bool"   and a0 :: "('f \<times> 'd) list"   and a1 :: "'d \<Rightarrow> 'e" assumes "\<And>uu_. P [] uu_"   and "\<And>x e h eval. P h eval \<Longrightarrow> P ((x, e) # h) eval" shows "P a0 a1"
Inr_Inl_False: fixes x :: "'f"   and y :: "'e" shows "(Inr x = Inl y) = False"
assms_3: shows "\<forall>E. Coplanar A B C E \<longrightarrow> Per E D P"
matrix_to_iarray_fst_snd_Gauss_Jordan_column_k_det_P: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and n :: "'b" assumes "i \<le> nrows A"   and "k < ncols A" shows "fst (snd (Gauss_Jordan_column_k_det_P (n, i, A) k)) = fst (snd (Gauss_Jordan_column_k_det_P_iarrays (n, i, matrix_to_iarray A) k))"
H_def: fixes A :: "'a set" shows "H A = map_pmf (\<lambda>f. Max (f ` A \<union> {0})) (SL A)"
wuntil_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "wuntil \<phi> \<psi> \<pi>l" obtains "\<psi> \<pi>l"   | "\<phi> \<pi>l" and "wuntil \<phi> \<psi> (map stl \<pi>l)"
iso_is_arr: fixes f :: "'a" assumes "R.iso f" shows "R.arr f"
C_def: shows "C = (\<lambda>(x, y). x \<inter> y) ` (Ca \<times> Cb)"
not_eq_extremum: fixes a :: "'c multiset" shows "(a \<noteq> {#}) = ({#} \<subset># a)"
assms: shows "vinsert a b \<in>\<^sub>\<circ> Vset \<alpha>"
Nil_tl: fixes xs :: "'b list" shows "([] = tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
negatively_invariant_complD: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.negatively_invariant f X (X - M)" shows "auto_ll_on_open.positively_invariant f X M"
PiE_UNIV: shows "UNIV \<rightarrow>\<^sub>E UNIV = UNIV"
length_insort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "length (linorder.insort_key less_eq f x xs) = Suc (length xs)"
dom_simp_4: shows "J.dom J.j1 = J.Zero"
AxTangentBase: fixes e :: "'a Point"   and cone :: "'a Cone" shows "pbasepoint (tangentPlane e cone) = e"
hop_count_zero_oip_dip_sip': fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg (\<lambda>m. rreq_rrep_sn m \<and> msg_zhops m) \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:4..PAodv-:5} \<union> {PRreq-:n |n. True} \<longrightarrow> hops \<xi> = 0 \<longrightarrow> oip \<xi> = sip \<xi>) \<and> (l \<in> {PAodv-:6..PAodv-:7} \<union> {PRrep-:n |n. True} \<longrightarrow> hops \<xi> = 0 \<longrightarrow> dip \<xi> = sip \<xi>))"
smods_0_2: fixes p :: "real poly" shows "smods p 0 = (if p = 0 then [] else [p])"
Acts_Join: fixes F :: "'c program"   and G :: "'c program" shows "Acts (F \<squnion> G) = Acts F \<union> Acts G"
ldrop_eq_LNil: fixes n :: "enat"   and xs :: "'b llist" shows "(ldrop n xs = <>) = (llength xs \<le> n)"
sum_permutations_compose_right: fixes q :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and f :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "q permutes S" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (p \<circ> q))"
spans_image: fixes V :: "'b set"   and B :: "'b set" assumes "V \<subseteq> m1.span B" shows "f ` V \<subseteq> m2.span (f ` B)"
intros_17: fixes x :: "name"   and b :: "name"   and P :: "pi" shows "<\<nu>x>x{b}.P \<equiv>\<^sub>e \<zero>"
g_C1_diff: shows "\<gamma> C1_differentiable_on {a..b} - k"
sup_left_divisibility: 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 "less_eq x y = (\<exists>z. sup x z = y)"
is_multi_arith_prog_on_def: fixes l :: "nat"   and m :: "nat"   and start :: "int"   and steps :: "nat \<Rightarrow> nat"   and a :: "int"   and b :: "int" shows "is_multi_arith_prog_on l m start steps a b = (a \<le> start \<and> multi_arith_prog m start steps (\<lambda>_. l - 1) \<le> b)"
assms: shows "rt \<noteq> update rt dip (osn, kno, val, hops, nhip, {})"
th: shows "\<forall>n. \<bar>Re (s n)\<bar> \<le> r + 1"
op_norm_le_transpose: fixes A :: "((real, 'a) vec, 'a) vec" shows "\<parallel>A\<parallel>\<^sub>o\<^sub>p \<le> \<parallel>Finite_Cartesian_Product.transpose A\<parallel>\<^sub>o\<^sub>p"
rel_mono_strong: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('a, 'c, 'e) gpv"   and y :: "('b, 'd, 'e) gpv"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_gpv R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> results'_gpv x; y1 \<in> results'_gpv y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> outs'_gpv x; y2 \<in> outs'_gpv y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_gpv R1a R2a x y"
leg1_commutes: shows "g.cod.leg1 \<cdot> g.chine = g.leg1"
divides_degree: fixes p :: "'c poly"   and q :: "'c poly" assumes "p dvd q" shows "degree p \<le> degree q \<or> q = 0"
set_update_by_pair_zero: fixes xs :: "('a \<times> 'c) list"   and k :: "'a" assumes "oalist_inv_raw xs" shows "set (update_by_pair (k, 0::'c) xs) = set xs - range (Pair k)"
below_in_P_simplex_like: fixes x :: "'a" assumes "x \<in> P" shows "simplex_like (P .\<^bold>\<le> x)"
completely_multiplicative_function'_def: fixes f :: "nat \<Rightarrow> 'b"   and f_prime :: "nat \<Rightarrow> 'b" shows "completely_multiplicative_function' f f_prime \<equiv> completely_multiplicative_function f \<and> completely_multiplicative_function'_axioms f f_prime"
cases: fixes x :: "('e, 'f) lit" obtains at :: "('e, 'f) atm" where "x = Pos at__"   | at :: "('e, 'f) atm" where "x = Neg at"
seqI: fixes f :: "J.Arr"   and g :: "J.Arr" assumes "J.arr f"   and "J.arr g"   and "J.dom g = J.cod f" shows "J.seq g f"
power_Suc: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
interchange_rows_transpose: fixes A :: "(('d, 'f) vec, 'e) vec"   and a :: "'f"   and b :: "'f" shows "interchange_rows (Finite_Cartesian_Product.transpose A) a b = Finite_Cartesian_Product.transpose (interchange_columns A a b)"
preserves_arr: fixes f :: "'a" assumes "R.arr f" shows "R.arr (R.map f)"
homeomorphic_contractible_eq: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "contractible S = contractible T"
error_free_check_method_access_eq: fixes G :: "prog"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and sig :: "sig"   and a' :: "val"   and s :: "state" assumes "error_free (check_method_access G accC statT mode sig a' s)" shows "check_method_access G accC statT mode sig a' s = s"
Tf_append: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" shows "Tf (t1 @ t2) <:: Tf t1 \<sqinter> Tf t2"
valid_x: shows "\<tau> \<Turnstile> \<upsilon> x"
is_valid_problem_sas_plus_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem" shows "is_valid_problem_sas_plus \<Psi> \<equiv> let ops = \<Psi>\<^sub>\<O>\<^sub>+; vs = \<Psi>\<^sub>\<V>\<^sub>+; I = \<Psi>\<^sub>I\<^sub>+; G = \<Psi>\<^sub>G\<^sub>+; D = range_of \<Psi> in list_all (\<lambda>v. D v \<noteq> None) vs \<and> list_all (is_valid_operator_sas_plus \<Psi>) ops \<and> (\<forall>v. (I v \<noteq> None) = ListMem v vs) \<and> (\<forall>v. I v \<noteq> None \<longrightarrow> ListMem (the (I v)) (the (D v))) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem v (\<Psi>\<^sub>\<V>\<^sub>+)) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem (the (G v)) (the (D v)))"
mem_case_prodI2: fixes p :: "'b \<times> 'c"   and z :: "'d"   and c :: "'b \<Rightarrow> 'c \<Rightarrow> 'd set" assumes "\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b" shows "z \<in> (case p of (a, b) \<Rightarrow> c a b)"
ev: shows "\<rho> \<turnstile> e \<Down> w"
FORK_refines_FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m: fixes N :: "nat"   and i :: "nat" assumes "DiningPhilosophers N" shows "DiningPhilosophers.FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m N i 0 \<sqsubseteq> DiningPhilosophers.FORK N i"
det_state_cover_card: fixes M :: "('d, 'e, 'f) FSM"   and V :: "'d list set" assumes "is_det_state_cover M V"   and "finite (nodes M)" shows "card (d_reachable M (initial M)) = card V"
of_int_divide_in_Ints: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "int"   and a :: "int" assumes "class.idom_divide times one plus zero minus uminus divide"   and "b dvd a" shows "divide (ring_1.of_int one plus zero minus a) (ring_1.of_int one plus zero minus b) \<in> ring_1.Ints one plus zero minus"
distinct_47: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "x31 \<oplus> x32 \<noteq> call(x10)"
ennreal_integral_real: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a pmf"   and B :: "real" assumes "measure_pmf.random_variable M borel f"   and "AE x in measure_pmf M. f x \<le> ennreal B"   and "0 \<le> B" shows "ennreal (measure_pmf.expectation M (\<lambda>x. enn2real (f x))) = integral\<^sup>N (measure_pmf M) f"
nb\<^sub>2: shows "cnf ` set (map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> \<^bold>\<not> (Atom (State (Suc t) (index (\<Pi>\<^sub>\<V>) v)))) (delete_effects_of op)) = (\<lambda>v. {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))\<inverse>}}) ` set (delete_effects_of op)"
simps_5: fixes S :: "'a formula \<Rightarrow> 'a formula"   and F :: "'a formula"   and G :: "'a formula" shows "freshnats.tseytin_tran1 S (F \<^bold>\<or> G) = [S (F \<^bold>\<or> G) \<^bold>\<leftrightarrow> S F \<^bold>\<or> S G] @ freshnats.tseytin_tran1 S F @ freshnats.tseytin_tran1 S G"
cont_Seq_finished: fixes c1 :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat"   and c2 :: "('test, 'atom, 'choice) cmd" assumes "finished (local.cont c1 s i)" shows "local.cont (c1 ;; c2) s i = c2"
vars_in_right_only: fixes x :: "nat"   and q :: "'a mpoly"   and p :: "'a mpoly" assumes "x \<in> vars q"   and "x \<notin> vars p" shows "x \<in> vars (p + q)"
NF_not_suc: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set" assumes "(x, y) \<in> r\<^sup>*"   and "x \<in> NF r" shows "x = y"
eq_statesI: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "\<R>\<^sub>G \<alpha> \<beta> s = s'"   and "bij \<alpha>"   and "bij \<beta>" shows "s \<approx> s'"
refine: fixes cnfi :: "(lit \<times> lit) list"   and cnf :: "lit uprod set" assumes "(cnfi, cnf) \<in> cnfi_rel" shows "(sati cnfi, satisfiable cnf) \<in> bool_rel"
1: shows "lselect s (ltake (enat k) w) = <>"
h2: assumes "\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)" shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
reachable_same_var: fixes \<phi>' :: "'val"   and g :: "'g"   and \<phi> :: "'val" assumes "\<phi>' \<in> reachable g \<phi>" shows "var g \<phi> = var g \<phi>'"
unique_Cons: fixes x :: "'c"   and y :: "'d"   and l :: "('c \<times> 'd) list" shows "unique ((x, y) # l) = (unique l \<and> (\<forall>y. (x, y) \<notin> set l))"
iEx_iAll_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t I. P t) = (\<not> (\<box> t I. \<not> P t))"
Carmichael_number_imp_squarefree_aux: fixes n :: "nat"   and p :: "nat"   and r :: "nat"   and l :: "nat" assumes "Carmichael_number n"   and "n = p ^ r * l"   and "prime p"   and "\<not> p dvd l"   and "1 < r" shows "False"
maximal_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "maximal t \<equiv> \<forall>c\<ge>0. t (\<lambda>_. c) = (\<lambda>_. c)"
unat_is_aligned_add: fixes p :: "'b word"   and n :: "nat"   and d :: "'b word" assumes "is_aligned p n"   and "unat d < 2 ^ n" shows "unat (p + d AND mask n) = unat d \<and> unat (p + d AND NOT (mask n)) = unat p"
oconf_upd_obj: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and obj :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and S' :: "(char list list \<times> (char list \<Rightarrow> val option)) set" assumes "P,h \<turnstile> obj \<surd>"   and "h a = \<lfloor>(C, S)\<rfloor>" shows "P,h(a \<mapsto> (C, S')) \<turnstile> obj \<surd>"
bitlen_alt_def: fixes a :: "int" shows "bitlen a = (if 0 < a then \<lfloor>log 2 (real_of_int a)\<rfloor> + 1 else 0)"
in_reg_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and curr_win :: "'b word"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = in_reg_mod w curr_win rd s1 \<and> t2 = in_reg_mod w curr_win rd s2" shows "low_equal t1 t2"
solve_def: fixes solve_exec :: "('a \<times> constraint) list \<Rightarrow> 'a list + (nat, rat) mapping"   and cs :: "('a \<times> constraint) list" assumes "SolveExec solve_exec" shows "SolveExec.solve solve_exec cs \<equiv> case solve_exec cs of Inl c \<Rightarrow> Inl c | Inr v \<Rightarrow> Inr \<langle>v\<rangle>"
inv_equality: fixes y :: "int"   and x :: "int" assumes "y \<otimes>\<^bsub>int_group\<^esub> x = \<one>\<^bsub>int_group\<^esub>"   and "x \<in> carrier int_group"   and "y \<in> carrier int_group" shows "inv\<^bsub>int_group\<^esub> x = y"
assms_1: fixes u :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>n. u n \<in> S"   and "b \<in> T"   and "u \<longlonglongrightarrow> b" shows "convergent (\<lambda>n. f (u n))"
Units_bij_betwI: fixes \<alpha> :: "'a \<Rightarrow> 'a" assumes "\<alpha> \<in> Sym" shows "bij_betw \<alpha> S S"
empty_is_image: fixes f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" shows "({} = (f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set)) = (A = {})"
infinite_growing: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'d set" assumes "class.linorder less_eq less"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. less x y" shows "infinite X"
Disj_I1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A" shows "H \<turnstile> A OR B"
inf_left_zero: fixes x :: "'a" shows "bot \<sqinter> x = bot"
Inf_utrans_lower: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" assumes "t \<in> S"   and "\<forall>t\<in>S. \<forall>P. unitary P \<longrightarrow> unitary (t P)" shows "le_utrans (Inf_utrans S) t"
en_eq_4: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1"   and y :: "\<kappa>"   and G :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace> \<^bold>\<equiv> (\<^bold>\<box>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>y,G\<rbrace>) in v]"
components_are_iso: fixes a :: "'b" assumes "ide a" shows "local.iso (\<rr>'.map a)"
\<mu>: shows "\<mu> = Cantor_sum \<alpha>s ms"
alphaAbs_preserves_qGoodAbs: fixes A :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and A' :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "A $= A'" shows "qGoodAbs A = qGoodAbs A'"
J_def: shows "J \<equiv> {uu_. \<exists>x X. uu_ = x \<and> (x \<in> V \<or> X \<in> XS \<and> \<not> qAFresh xs x X)}"
lossless_restrict_gpvD: fixes gpv :: "('b, 'out, 'in) gpv" assumes "lossless_gpv \<I> (local.restrict_gpv gpv)"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_gpv \<I> gpv"
runit'_in_vhom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "src f = a" shows "partial_magma.in_hom V (category.inv V (bicategory.runit V H \<a> \<i> src f)) f (H f a)"
polyn_expr_split: fixes k :: "nat"   and f :: "nat \<times> (nat \<Rightarrow> 'a)" shows "polyn_expr R X k f = polyn_expr R X k (fst f, snd f)"