hcju/isabelleps huggingface dataset:
lexext_rev_simps_2: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "\<not> lexext_rev gt [] xs"
d_OUT_mono: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>y. f (x, y) \<le> g (x, y)" shows "d_OUT f x \<le> d_OUT g x"
D_NEW: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and C :: "Class" assumes "CONTEXT x = low" shows "(G, New x C, Sec (\<lambda>(s, t, \<beta>). \<exists>l r. l \<notin> Dom (snd r) \<and> r \<equiv>\<^sub>\<beta> t \<and> s = (update (fst r) x (RVal (Loc l)), (l, C, []) # snd r))) \<in> Deriv"
set_tag_name_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_tag_name_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
cases: fixes x :: "'a \<times> ('b, 'c) antecedent" obtains uu_ :: "'a" and p :: "('b, 'c) antecedent" where "x = (uu_, p)"
i_not_k2: shows "from_nat k \<noteq> from_nat i"
mset_subseteq_add_iff1: fixes j :: "nat"   and i :: "nat"   and u :: "'k multiset"   and m :: "'k multiset"   and n :: "'k multiset" assumes "j \<le> i" shows "(repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (repeat_mset (i - j) u + m \<subseteq># n)"
has_derivative_split_at: fixes g :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and x :: "real" assumes "(g has_derivative g') (at_left x)"   and "(g has_derivative g') (at_right x)" shows "(g has_derivative g') (at x)"
him: shows "h ` \<Union> \<H> \<subseteq> rel_frontier T"
mapi_branch_mem: fixes ps :: "('b, 'c) fm list"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and f :: "nat \<Rightarrow> nat \<Rightarrow> ('b, 'c) fm \<Rightarrow> ('b, 'c) fm" assumes "(ps, i) \<in>. branch" shows "\<exists>v. (mapi (f v) ps, i) \<in>. mapi_branch f branch"
uminus_closed: fixes x :: "'a" assumes "x \<in> A" shows "- x \<in> A"
two_pred_none: fixes c :: "one\<^sub>\<bottom>" shows "(two_pred\<cdot>c = none) = (c \<sqsubseteq> once)"
steps_append: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph_Defs.steps E xs"   and "Graph_Defs.steps E ys"   and "last xs = hd ys" shows "Graph_Defs.steps E (xs @ tl ys)"
rbd_disj2: fixes R :: "('c \<times> 'd) set" shows "sup_pres (bd\<^sub>\<R> R)"
set_list_of_B: fixes t :: "'b tree" assumes "braun t" shows "set (list_of_B t) = set_tree t"
I_cooper1_hd_coeffs1: fixes \<phi> :: "atom fm"   and xs :: "int list" assumes "Z.normal \<phi>"   and "nqfree \<phi>" shows "Z.I (qe_cooper\<^sub>1 (hd_coeffs1 \<phi>)) xs = (\<exists>x. Z.I \<phi> (x # xs))"
min_zpassign_simps_1: fixes x :: "'v" shows "zhmset_of (min_passign (PWt x)) = zhmset_of (wt_sym (min_ground_head (Var x)))"
girth_def: fixes G :: "nat set \<times> nat set set" shows "girth G \<equiv> \<Sqinter>p\<in>ucycles G. enat (uwalk_length p)"
continuous_apply: fixes f :: "'b set \<Rightarrow> 'c set"   and C :: "nat \<Rightarrow> 'b set" assumes "continuous f"   and "chain C" shows "f (natUnion C) = natUnion (f \<circ> C)"
construct_rhs_vector_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and Is :: "(nat list \<times> nat list) list" shows "construct_rhs_vector_R p qs Is = vec_of_list (map (\<lambda>(I1, I2). construct_NofI_R p (retrieve_polys qs I1) (retrieve_polys qs I2)) Is)"
null_def: shows "null = (THE n. \<forall>f. local.comp n f = n \<and> local.comp f n = n)"
simps_2: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_ZObis tval aval indis (c1 ;; c2) = (PL_Indis.SC_siso tval aval indis c1 \<and> PL_Indis.SC_ZObis tval aval indis c2 \<or> PL_Indis.SC_ZObis tval aval indis c1 \<and> PL_Indis.SC_discr aval indis c2 \<or> PL_Indis.SC_discr aval indis (c1 ;; c2) \<or> PL_Indis.SC_siso tval aval indis (c1 ;; c2))"
Gamma_real_altdef1: fixes x :: "real" shows "Gamma x = lim (Gamma_series x)"
vfsequence_vlrestriction_succ: fixes xs :: "V"   and k :: "V" assumes "vfsequence xs"   and "k \<in>\<^sub>\<circ> vcard xs" shows "xs \<restriction>\<^sup>l\<^sub>\<circ> ZFC_in_HOL.succ k = xs \<restriction>\<^sup>l\<^sub>\<circ> k #\<^sub>\<circ> xs\<lparr>k\<rparr>"
delta_mono_def: fixes x :: "'c" shows "delta_mono x \<equiv> (1::'c) \<le> x"
elims: fixes x :: "'c tree"   and y :: "nat" assumes "height x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'c tree" and a :: "'c" and r :: "'c tree" where "x = \<langle>l, a, r\<rangle>" and "y = max (height l) (height r) + 1"
cimage_def: shows "(`\<^sub>c) \<equiv> map_fun id (map_fun rcset acset) (`)"
ereal_complete_uminus_eq: fixes S :: "ereal set"   and x :: "ereal" shows "((\<forall>y\<in>uminus ` S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus ` S. y \<le> z) \<longrightarrow> x \<le> z)) = ((\<forall>y\<in>S. - x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> - x))"
times_set_def: fixes A :: "'a set"   and B :: "'a set" shows "pseudo_hoop_algebra.times_set (\<lambda>a b. b * a) A B = {a. \<exists>x\<in>A. \<exists>y\<in>B. a = y * x}"
wordinterval_CIDR_split_prefixmatch_all_valid_Ball: fixes r :: "'a wordinterval" shows "\<forall>e\<in>set (wordinterval_CIDR_split_prefixmatch r). valid_prefix e \<and> pfxm_length e \<le> LENGTH('a)"
pos_divide_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less (divide b c) a = less b (times a c)"
Sigma_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd set"   and D :: "'c \<Rightarrow> 'd set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
trivial_group_def: fixes G :: "('b, 'c) monoid_scheme" shows "trivial_group G \<equiv> Group.group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
fcs_x_eq_x: fixes x :: "'b" shows "(G.face_cycle_succ x = x) = (G.face_cycle_set x = {x})"
simps_3: fixes x31a :: "'b local.term"   and x32a :: "'b local.term"   and y31a :: "'b local.term"   and y32a :: "'b local.term" shows "(x31a \<^bold>\<star> x32a = y31a \<^bold>\<star> y32a) = (x31a = y31a \<and> x32a = y32a)"
subseq_reduce: fixes s :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono s" shows "strict_mono (reduce s n)"
reaches_steps_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>* y) = (\<exists>xs. hd xs = x \<and> last xs = y \<and> steps xs)"
type_CONJ: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "inv ni_bij \<circ> f \<circ> ni_bij \<in> S_inf"
assms_4: shows "e ; s\<^sup>T \<le> x"
prems: shows "x \<in> vars_of (t1 \<cdot> t2)"
finite_measure_subalgebra_def: fixes M :: "'b measure"   and F :: "'b measure" shows "finite_measure_subalgebra M F \<equiv> finite_measure M \<and> finite_measure_subalgebra_axioms M F"
rank_invar_hd_cons: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank_invar bq"   and "SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank (hd bq)" shows "SkewBinomialHeapStruc.rank_invar (t # bq)"
induct: fixes P :: "('a \<times> 'b \<Rightarrow> 'a \<times> 'c) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> bool"   and a0 :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c"   and a1 :: "('a \<times> 'b) list" assumes "\<And>f. P f []"   and "\<And>f kv xs. (\<And>x xa y. \<lbrakk>x = f kv; (xa, y) = x\<rbrakk> \<Longrightarrow> P f xs) \<Longrightarrow> P f (kv # xs)" shows "P a0 a1"
AIjeq: shows "\<Gamma> ! j = (p \<leftrightarrow> q)"
monoidal_category_axioms: shows "monoidal_category (\<cdot>) T\<^sub>F\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>M\<^sub>C"
dist_fls_def: fixes a :: "'a fls"   and b :: "'a fls" shows "dist a b = (if a = b then 0 else if 0 \<le> fls_subdegree (a - b) then inverse (2 ^ nat (fls_subdegree (a - b))) else 2 ^ nat (- fls_subdegree (a - b)))"
quorum_of_set_def: fixes S :: "'node set"   and Q :: "'node set" shows "quorum_of_set S Q \<equiv> \<exists>p\<in>S. quorum_of p Q"
eq: shows "subgroup_generated (reduced_homology_group (int p) (nsphere p)) {a} = reduced_homology_group (int p) (nsphere p)"
one_in_Polys: fixes X :: "'c set" shows "1 \<in> P[X]"
size_of_lhs_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "dim_vec (w_vec_R p qs signs) = length signs"
Bclo: fixes i :: "'m"   and j :: "'n" shows "\<bar>B $ i $ j - A $ i $ j\<bar> < e / real (2 * CARD('m) * CARD('n))"
inject_1: fixes x1 :: "int"   and y1 :: "int" shows "(Num x1 = Num y1) = (x1 = y1)"
Rel_Rel_match: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f" assumes "Transfer.Rel R x y" shows "Rel_match R x y"
set_lebesgue_integral_cong: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "A \<in> sets M"   and "\<forall>x. x \<in> A \<longrightarrow> f x = g x" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A g"
bex_nonempty_set: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" shows "{x \<in> A. P x} \<noteq> {}"
sup_parity_def: fixes x :: "parity"   and y :: "parity" shows "x \<squnion> y = (if x = y then x else Either)"
nn_pp_np_2: fixes f :: "'b \<Rightarrow> 'c" assumes "nonnegative f" shows "np f = (\<lambda>t. 0::'c)"
ACI_Plus_assocI_2: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and s2 :: "'b rexp"   and s1 :: "'b rexp" assumes "ACI (Plus r1 r2) s2" shows "ACI (Plus r1 (Plus r2 s1)) (Plus s1 s2)"
lemma1b: fixes tp :: "bool list"   and tq :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" assumes "(tp, tq) \<in> (eps (or L R))\<^sup>*"   and "tp = False # p" shows "\<exists>q. (p, q) \<in> (eps R)\<^sup>* \<and> tq = False # q"
sr_imp_eigen_vector: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and u :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A"   and "u \<in> perron_frobenius.X"   and "perron_frobenius.r A u = perron_frobenius.r A (perron_frobenius.z A)" shows "eigen_vector A u (perron_frobenius.r A (perron_frobenius.z A))"
circuit_subset_eq: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and C\<^sub>1 :: "'b set"   and C\<^sub>2 :: "'b set" assumes "indep_system carrier indep"   and "indep_system.circuit carrier indep C\<^sub>1"   and "indep_system.circuit carrier indep C\<^sub>2"   and "C\<^sub>1 \<subseteq> C\<^sub>2" shows "C\<^sub>1 = C\<^sub>2"
consec_lesser: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> (\<forall>n m. n \<^bold>\<in> i \<and> m \<^bold>\<in> j \<longrightarrow> n < m)"
oa_facts_2: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
kop_galois_var2: fixes f :: "('b \<Rightarrow> 'a set) \<Rightarrow> 'c"   and g :: "('a \<Rightarrow> 'b set) \<Rightarrow> 'c" shows "(f \<circ> op\<^sub>K = g) = (g \<circ> op\<^sub>K = f)"
alpha_Tree_reflp: shows "reflp (=\<^sub>\<alpha>)"
unique_session_keys: fixes A :: "agent"   and K :: "nat"   and B :: "agent"   and N :: "msg"   and RB :: "msg"   and A' :: "agent"   and B' :: "agent"   and N' :: "msg"   and PB :: "msg"   and KXY :: "nat"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key K, Agent B, N\<rbrace> \<in> parts {RB}"   and "Crypt (shrK A') \<lbrace>Key K, Agent B', N'\<rbrace> \<in> parts {RB}"   and "(PB, RB, KXY) \<in> respond evs" shows "A' = A \<and> B' = B \<or> A' = B \<and> B' = A"
comp_char: fixes g :: "'a"   and f :: "'a" shows "Right_a.comp g f = (if Right_a.arr f \<and> Right_a.arr g \<and> seq g f then g \<cdot> f else null)"
simps_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x1 :: "'c" shows "(case Pos x1 of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = f1 x1"
l_zero: fixes x :: "'c" assumes "x \<in> carrier S" shows "\<zero>\<^bsub>S\<^esub> \<oplus>\<^bsub>S\<^esub> x = x"
dependent_imp_affine_dependent: fixes a :: "'a"   and s :: "'a set" assumes "dependent {x - a |x. x \<in> s}"   and "a \<notin> s" shows "affine_dependent (insert a s)"
lift_def: fixes A :: "nat set" shows "lift A \<equiv> insert 0 (Suc ` A)"
PiE_cong: fixes I :: "'f set"   and A :: "'f \<Rightarrow> 'g set"   and B :: "'f \<Rightarrow> 'g set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
nearly_healthy_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "nearly_healthy t = ((\<forall>P. unitary P \<longrightarrow> unitary (t P)) \<and> (\<forall>P Q. unitary P \<longrightarrow> unitary Q \<longrightarrow> P \<tturnstile> Q \<longrightarrow> t P \<tturnstile> t Q))"
conj_cancel_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x (compl x) = zero"
Ex_condensation_leaf: fixes P :: "'val set"   and g :: "'g" assumes "P \<noteq> {}" shows "\<exists>leaf. leaf \<in> condensation_nodes g P \<and> (\<forall>scc. (leaf, scc) \<notin> condensation_edges g P)"
fls_base_factor_to_fps_trivial: fixes f :: "'a fls" assumes "fls_subdegree f = 0" shows "fls_base_factor_to_fps f = fls_regpart f"
nprv_addDsjLemmaE: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "prv (dsj \<phi>1 \<phi>2)"   and "nprv (insert \<phi>1 F) \<psi>"   and "nprv (insert \<phi>2 F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
case_prod_beta: fixes f :: "'m \<Rightarrow> 'n \<Rightarrow> 'l"   and p :: "'m \<times> 'n" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
cod_lunit: fixes a :: "'d" assumes "D.ide a" shows "D.cod \<l>\<^sub>1[a] = a"
map_dual_inv_def: fixes f :: "'c dual \<Rightarrow> 'd dual" shows "map_dual_inv f = \<partial>\<^sup>- \<circ> f \<circ> \<partial>"
path_iff_aux_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and star :: "'b \<Rightarrow> 'b"   and r :: "'b"   and x :: "'b" assumes "class.relation_algebra_rtc minus uminus inf less_eq less sup bot top composition converse unit star"   and "relation_algebra.is_inj less_eq composition converse unit r \<and> relation_algebra.is_sur less_eq composition converse unit r" shows "less_eq (composition r x) (sup (star x) (star (converse x))) = less_eq x (composition (converse r) (sup (star x) (star (converse x))))"
1: fixes i :: "nat"   and j :: "nat" shows "P (iterate i\<cdot>f\<cdot>\<bottom>) (iterate j\<cdot>g\<cdot>\<bottom>) \<and> Q (iterate i\<cdot>f\<cdot>\<bottom>) (iterate j\<cdot>g\<cdot>\<bottom>)"
connected_component_in: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "connected_component S x y" shows "x \<in> S \<and> y \<in> S"
invar_filter: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and f :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree \<Rightarrow> bool" assumes "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_invar (filter f q)"
hyp2_axiom2: shows "\<forall>a b p q r s. a b \<congruent>\<^sub>K p q \<and> a b \<congruent>\<^sub>K r s \<longrightarrow> p q \<congruent>\<^sub>K r s"
distinct_signs_hyp: shows "distinct (snd (snd (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))))"
lnth_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'c llist" assumes "enat (n + m) < llength xs" shows "lnth (ldropn n xs) m = lnth xs (m + n)"
plus_class: fixes x :: "'b"   and y :: "'b" shows "x \<oplus>\<^bsub>cring_class_ops\<^esub> y = x + y"
permutations_of_set_infinite: fixes A :: "'a set" assumes "infinite A" shows "permutations_of_set A = {}"
cring_simprules_7: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
orthogonal: fixes xa :: "nat" assumes "xa < i" shows "gso i \<bullet> gso xa = (0::'a)"
converseI: fixes a :: "'h"   and b :: "'i"   and r :: "('h \<times> 'i) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
uses_in_allUses: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> uses g n" shows "v \<in> allUses g n"
independent_explicit: fixes B :: "'b set" shows "independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *s v) = (0::'b) \<longrightarrow> (\<forall>v\<in>B. c v = (0::'a))))"
inversions_between_permute: fixes \<pi>1 :: "nat \<Rightarrow> nat"   and xs :: "'a list"   and \<pi>2 :: "nat \<Rightarrow> nat"   and ys :: "'a list" assumes "\<pi>1 permutes {..<length xs}"   and "\<pi>2 permutes {..<length ys}" shows "inversions_between (permute_list \<pi>1 xs) (permute_list \<pi>2 ys) = map_prod (inv \<pi>1) (inv \<pi>2) ` inversions_between xs ys"
flow_undefined_2: fixes x0 :: "'a"   and t0 :: "real" assumes "x0 \<notin> X" shows "flow t0 x0 = (\<lambda>_. 0::'a)"
subprob_cexprI: fixes V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and V :: "nat set"   and e :: "cexpr" assumes "\<And>\<rho>. \<rho> \<in> space (state_measure V' \<Gamma>) \<Longrightarrow> \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge V V' (x, \<rho>)) e)) \<partial>state_measure V \<Gamma> \<le> 1" shows "subprob_cexpr V V' \<Gamma> e"
connected_edgeRtc: fixes u :: "nat"   and v :: "nat" shows "local.connected u v = ((u, v) \<in> E\<^sup>*)"
rep_eq: fixes x :: "mini_alg_unique"   and xa :: "mini_alg_unique" shows "mau_compatible x xa = ma_compatible (Rep_mini_alg_unique x) (Rep_mini_alg_unique xa)"
exteq_trans: fixes f :: "recf"   and g :: "recf"   and h :: "recf" assumes "f \<simeq> g"   and "g \<simeq> h" shows "f \<simeq> h"
Q_zip_weak: fixes z :: "('a \<times> 'c) H" assumes "Q (map_H fst z)"   and "Q (map_H snd z)" shows "\<exists>z'. Q z' \<and> set_H z' \<subseteq> set_H z \<and> map_H fst z' = map_H fst z \<and> map_H snd z' = map_H snd z"
measurable_completion: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> completion M \<rightarrow>\<^sub>M N"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>\<sqinter> b = b \<^bold>\<sqinter> a"
Int_ac_3: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B = B \<inter> A"
dm: fixes dm :: "qtname \<times> methd" assumes "methd G declC sig = Some dm"   and "dynlookup G statT invC sig = Some dm"   and "G\<turnstile>resTy (mthd dm)\<preceq>resTy sm"   and "wf_mdecl G declC (sig, mthd dm)"   and "declC = declclass dm"   and "is_static dm = is_static sm"   and "is_class G invC"   and "is_class G declC"   and "G\<turnstile>invC\<preceq>\<^sub>C declC"   and "if invmode sm e = IntVir then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC) \<or> (\<forall>statC. statT \<noteq> ClassT statC \<and> declC = Object)) \<and> statDeclT = ClassT (declclass dm)" shows "P"
ftx: shows "(\<lambda>x. f (t + x)) absolutely_integrable_on {- pi..pi}"
s_is_partial_execution: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "distributed_system channel" shows "distributed_system.s channel trans send recv c t i = distributed_system.run_trace channel trans send recv (c, take i t)"
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> ('a, 'c) rbt \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "'a"   and a2 :: "'c"   and a3 :: "('a, 'c) rbt" assumes "\<And>f k v. P f k v Empty"   and "\<And>f k v l x y r. \<lbrakk>k < x \<Longrightarrow> P f k v l; \<lbrakk>\<not> k < x; x < k\<rbrakk> \<Longrightarrow> P f k v r\<rbrakk> \<Longrightarrow> P f k v (Branch B l x y r)"   and "\<And>f k v l x y r. \<lbrakk>k < x \<Longrightarrow> P f k v l; \<lbrakk>\<not> k < x; x < k\<rbrakk> \<Longrightarrow> P f k v r\<rbrakk> \<Longrightarrow> P f k v (Branch R l x y r)" shows "P a0 a1 a2 a3"
lfp_loop_unitary: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "unitary P" shows "unitary (lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s))"
listAll2_map2I: fixes xs :: "'a list"   and ys :: "'b list"   and R :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> R (xs ! i) (f (xs ! i) (ys ! i))" shows "list_all2 R xs (map2 f xs ys)"
lcoset_perm_def: fixes e :: "'g set"   and g :: "'g set" assumes "Group e \<and> order.greater_eq G e"   and "normal e" shows "BinOpSetGroup.Abs_G_perm (G // LCoset_rel e) (+) g \<equiv> Abs_permutation (restrict1 ((+) g) (G // LCoset_rel e))"
measure_UN_bound: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "Sigma_Algebra.measure M (\<Union> (A ` I)) \<le> B"
that: fixes g :: "real \<Rightarrow> real" assumes "g square_integrable S"   and "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x)) \<longlonglongrightarrow> 0" shows "thesis"