hcju/isabelleps huggingface dataset:
intros_6: fixes dims :: "nat list"   and P :: "complex mat"   and Q :: "complex mat"   and P' :: "complex mat"   and Q' :: "complex mat"   and S :: "com" assumes "state_sig.is_quantum_predicate dims P"   and "state_sig.is_quantum_predicate dims Q"   and "state_sig.is_quantum_predicate dims P'"   and "state_sig.is_quantum_predicate dims Q'"   and "P \<le>\<^sub>L P'"   and "state_sig.hoare_partial dims P' S Q'"   and "Q' \<le>\<^sub>L Q" shows "state_sig.hoare_partial dims P S Q"
retraction_def: fixes g :: "'d" shows "Fop.Aop.retraction g \<equiv> \<exists>f. Fop.Aop.ide (Fop.Aop.comp g f)"
linear_strict_order_order: fixes x :: "'a" assumes "linear_orderable_2 x" shows "linear_orderable_1 (x \<squnion> (1::'a))"
ereal_mult_le_mult_iff: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "\<bar>c\<bar> \<noteq> \<infinity>" shows "(c * a \<le> c * b) = ((0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a))"
less_asym: fixes x :: "'a"   and y :: "'a"   and P :: "bool" assumes "x \<prec> y"   and "\<not> P \<Longrightarrow> y \<prec> x" shows "P"
valid_def: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and guru :: "'b"   and w :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "Blue_Eyes.valid blue guru w = (\<exists>p. p \<noteq> guru \<and> w p = blue)"
\<rho>: shows "\<rho> \<in> space (state_measure V' \<Gamma>)"
synthetic_div_pCons: fixes a :: "'b"   and p :: "'b poly"   and c :: "'b" shows "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
assms_2: shows "list.set (\<alpha> # \<alpha>s) \<subseteq> ON"
tl_stream_enc: fixes w :: "'a list"   and x :: "nat + nat set"   and I :: "(nat + nat set) list" shows "smap \<pi> (stream_enc (w, x # I)) = stream_enc (w, I)"
ur': shows "\<exists>!x. root_cond (q, l', r') x"
mult_eSuc: fixes m :: "enat"   and n :: "enat" shows "eSuc m * n = n + m * n"
taut_test: fixes b :: "'d bool_expr" shows "taut_test ifex_of b = (\<forall>s. val_bool_expr b s)"
invertible_imp_matrix_inv: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "invertible A" shows "matrix_inv A = (1::'a) div det A *k adjugate A"
assms: shows "summable (\<lambda>n. a n * complex_of_real (r ^ n))"
trans2: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and t :: "'a"   and r :: "'a"   and q :: "'a" assumes "class.interval meets \<I>"   and "meets p t"   and "meets t r"   and "meets r q" shows "\<not> meets p q"
fmdrop_fset_empty: fixes A :: "'a fset" shows "fmdrop_fset A fmempty = fmempty"
Domain_converse: fixes r :: "('b \<times> 'a) set" shows "Domain (r\<inverse>) = Range r"
rtrancl_last_touch: fixes q :: "'b"   and q' :: "'b"   and R :: "('b \<times> 'b) set"   and S :: "'b set" assumes "(q, q') \<in> R\<^sup>*"   and "q \<in> S" obtains qt :: "'b" where "qt \<in> S" and "(q, qt) \<in> R\<^sup>*" and "(qt, q') \<in> (R - UNIV \<times> S)\<^sup>*"
assms_2: shows "eval c t t' n (\<box>\<^sub>b\<gamma>)"
gcd_dvdI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "gcd a b dvd c"
res': shows "finite_field_factorization_i p ff_ops (of_int_poly_i ff_ops (Mp f)) = (c', fs')"
apex_def: shows "src.cod.apex \<equiv> C.dom src.cod.leg0"
ndet_symm: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and Q :: "'c \<times> 'c \<Rightarrow> bool" shows "(P \<or> Q) = (Q \<or> P)"
Phi_def: fixes n :: "nat" shows "\<Phi> n = int (card (Inv (s_A n) (s_mtf n)))"
takefill_bintrunc: fixes n :: "nat"   and bl :: "bool list" shows "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))"
order_iff: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (a = a \<sqinter> b)"
degree_pderiv: fixes p :: "'a poly" shows "Polynomial.degree (pderiv p) = Polynomial.degree p - 1"
inject_3: fixes x31 :: "nat"   and x32 :: "'e"   and y31 :: "nat"   and y32 :: "'e" shows "(unicast x31 x32 = unicast y31 y32) = (x31 = y31 \<and> x32 = y32)"
extend_Subspace_basis: fixes U :: "'b set"   and us :: "'b list" assumes "U \<subseteq> V"   and "set us \<subseteq> U \<and> U = RSpan us \<and> R_lin_independent us" shows "\<exists>vs. set (vs @ us) \<subseteq> V \<and> V = RSpan (vs @ us) \<and> R_lin_independent (vs @ us)"
simps_4: shows "eval_ternary_Imp TernaryFalse TernaryFalse = TernaryTrue"
add_nrows: fixes A :: "'a matrix"   and u :: "nat"   and B :: "'a matrix" assumes "nrows A \<le> u"   and "nrows B \<le> u" shows "nrows (A + B) \<le> u"
assms_4: shows "prv (imp \<phi> (neg \<phi>2))"
list_ConsCons_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
tensor_non_empty: fixes A :: "int poly fract list list"   and B :: "int poly fract list list" assumes "A \<noteq> []"   and "B \<noteq> []" shows "A \<otimes> B \<noteq> []"
matroid_ax_2: fixes X :: "Points set"   and Y :: "Points set" shows "X \<subseteq> Y \<longrightarrow> rk X \<le> rk Y"
emb_prod_def: shows "EMB('e \<times> 'f) = prod_emb oo prod_map\<cdot>EMB('e)\<cdot>EMB('f)"
reads_bind_pure: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l"   and S :: "('l \<Rightarrow> 'l \<Rightarrow> bool) set"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog" assumes "pure f h"   and "pure f h'"   and "reads S f h h'"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> reads S (g x) h h'" shows "reads S (Heap_Error_Monad.bind f g) h h'"
abs: fixes s :: "dB"   and t :: "dB" assumes "s => t" shows "Abs s => Abs t"
odd_iff: fixes B :: "nat set"   and x :: "nat" assumes "finite B" shows "odd (\<Sum>i\<in>B. if i < x then 0 else 2 ^ (i - x)) = (x \<in> B)"
int_fab: shows "(\<lambda>x. integral {x..b} f) ` {a..b} = {m..M}"
parents_no_parent: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and dir :: "dir"   and p :: "(nat \<times> int) list" assumes "d < length b" shows "b \<notin> parents d (child b dir d) p"
size_mset_union: fixes M :: "('d \<times> 'e) multiset"   and N :: "('d \<times> 'e) multiset" shows "size_mset (M + N) = size_mset N + size_mset M"
assms_3: shows "time (Rep_run r k c) = \<tau>"
siso_Sbis: fixes c :: "('test, 'atom, 'choice) cmd" assumes "siso c" shows "c \<approx>s c"
inject_1: fixes x11 :: "nat"   and x12 :: "nat"   and y11 :: "nat"   and y12 :: "nat" shows "(Less x11 x12 = Less y11 y12) = (x11 = y11 \<and> x12 = y12)"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "d_OUT f x = 0" shows "SINKp f x"
B'_2: fixes x :: "real" assumes "x \<in> {B'<..<B}" shows "norm (f x) \<le> c * norm (g x)"
isomorphic_categories_axioms: shows "isomorphic_categories C' (\<cdot>)"
not_all_zero: shows "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))"
tagged_fun_relD_none: fixes f :: "'g \<Rightarrow> 'h"   and f' :: "'i \<Rightarrow> 'j"   and A :: "('g \<times> 'i) set"   and B :: "('h \<times> 'j) set"   and x :: "'g"   and x' :: "'i" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.R f) = B.R (B.dom f)"
root_int_main_3: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool" assumes "0 \<le> n"   and "root_int_main p n = (y, b)"   and "0 < p" shows "y ^ p \<le> n"
ordered_sel_resolvent_def: fixes Sel :: "'at set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "ordered_sel_resolvent Sel P1 P2 C \<equiv> \<exists>A. C = P1 - {Pos A} \<union> (P2 - {Neg A}) \<and> strictly_maximal_literal P1 (Pos A) \<and> selected_part Sel P1 = {} \<and> (strictly_maximal_literal P2 (Neg A) \<and> selected_part Sel P2 = {} \<or> strictly_maximal_literal (selected_part Sel P2) (Neg A))"
is_functor: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag" assumes "identity_functor C" shows "functor C C (identity_functor.map C)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_locate_a (as, lm) (s, l, r) ires = (at_begin_norm (as, lm) (s, l, r) ires \<or> at_begin_fst_bwtn (as, lm) (s, l, r) ires \<or> at_begin_fst_awtn (as, lm) (s, l, r) ires)"
rel_cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d stream"   and b :: "'e stream" assumes "stream_all2 R a b" obtains x1a :: "'d" and x2a :: "'d stream" and y1a :: "'e" and y2a :: "'e stream" where "a = x1a ## x2a" and "b = y1a ## y2a" and "R x1a y1a" and "stream_all2 R x2a y2a"
Gauss_Jordan_column_k_iarrays_def: fixes A' :: "nat \<times> 'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_iarrays A' k = (let A = snd A'; i = fst A' in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (i, A) else (Suc i, Gauss_Jordan_in_ij_iarrays A i k))"
inf_mult_zero: shows "\<infinity>\<^sub>h *\<^sub>h 0\<^sub>h = 1\<^sub>h"
simps_1: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "RDocument.nothing \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = nothinga"
assms: shows "set TI' = {(a, b). (a, b) \<in> TI\<^sup>+ \<and> a \<noteq> b}"
hyps_3: fixes \<Gamma>2 :: "(name \<times> ty) list" assumes "valid \<Gamma>2"   and "(x, T1) # \<Gamma>1 \<subseteq> \<Gamma>2" shows "\<Gamma>2 \<turnstile> t : T2"
closed_scaling: fixes S :: "'d set"   and c :: "real" assumes "closed S" shows "closed ((*\<^sub>R) c ` S)"
Ints_add: 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 a :: "'b"   and b :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "plus a b \<in> ring_1.Ints one plus zero minus"
coprime_mod_right_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "a \<noteq> zero" shows "algebraic_semidom.coprime times one a (modulo b a) = algebraic_semidom.coprime times one a b"
view_determination: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and u :: "'e"   and \<rho> :: "'f"   and v :: "'e" assumes "weak_lens x"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "u = v"
length_inorder: fixes t :: "'b tree" shows "length (inorder t) = size t"
conjugate_smult_vec: fixes k :: "'b"   and v :: "'b vec" shows "conjugate (k \<cdot>\<^sub>v v) = conjugate k \<cdot>\<^sub>v conjugate v"
t_init: shows "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd TNs)"
select_convs_2: fixes orderA :: "('f, 'g) gorder_scheme"   and orderB :: "('d, 'e) gorder_scheme"   and lower :: "'f \<Rightarrow> 'd"   and upper :: "'d \<Rightarrow> 'f"   and more :: "'h" shows "\<Y>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderB"
all1_matrix_def: fixes nr :: "nat"   and nc :: "nat" shows "all1_matrix nr nc = mat nr nc (\<lambda>(r, c). 1)"
alpha_tConj: fixes tset1 :: "('e, 'f, 'g) Tree set['e]"   and tset2 :: "('e, 'f, 'g) Tree set['e]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
fset_inverse: fixes x :: "'c fset" shows "Abs_fset (fset x) = x"
ge: shows "poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'a), PVar v] else PVar w)) p \<ge>p p"
fps_cos_fps_exp_ii: fixes c :: "complex" shows "fps_cos c = (fps_exp (\<i> * c) + fps_exp (- \<i> * c)) / fps_const 2"
add_le_same_cancel1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
set_iterator_abs_rule_P: assumes "set_iterator_abs \<alpha> invar it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; invar x; \<alpha> x \<in> S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (S - {\<alpha> x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> True; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
mcont_bind_spmf1: fixes f :: "'c \<Rightarrow> 'd spmf" shows "mcont lub_spmf (ord_spmf (=)) lub_spmf (ord_spmf (=)) (\<lambda>y. y \<bind> f)"
has_derivative_subset: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "(f has_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_derivative f') (at x within t)"
wf_theory_imp_is_std_sig: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "wf_theory \<Theta>" shows "is_std_sig (sig \<Theta>)"
iso': shows "restrict (inv_into (carrier G1) i) (carrier G2) \<in> Group.iso G2 G1"
lebesgue_integrable_real_affine: fixes f :: "real \<Rightarrow> 'c"   and c :: "real"   and t :: "real" assumes "integrable lebesgue f"   and "c \<noteq> 0" shows "integrable lebesgue (\<lambda>x. f (t + c * x))"
k_isNode_cong: fixes ec' :: "'rep"   and ec :: "'rep" assumes "simAbs ec' = simAbs ec" shows "k_isNode ec' = k_isNode ec"
solve_depressed_quartic_complex_def: fixes p :: "complex"   and q :: "complex"   and r :: "complex" shows "solve_depressed_quartic_complex p q r = remdups (if q = 0 then concat (map (\<lambda>z. let y = csqrt z in [y, - y]) (croots2 [:r, p, 1:])) else let cubics = Cubic_Polynomials.croots3 [:- q\<^sup>2, 2 * p\<^sup>2 - 8 * r, 8 * p, 8:]; m = hd cubics; a = csqrt (2 * m); p2m = p / 2 + m; q2a = q / (2 * a); b1 = p2m - q2a; b2 = p2m + q2a in croots2 [:b1, a, 1:] @ croots2 [:b2, - a, 1:])"
map_uprod_simps: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'c" shows "map_uprod f (Upair x y) = Upair (f x) (f y)"
set_child_nodes_locs_def: shows "set_child_nodes_locs = (\<lambda>ptr. (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes_update) else {}) \<union> (if is_document_ptr_kind ptr then all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element_update) else {}))"
CatchThrow: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile> (Catch Throw c\<^sub>2, Normal s) \<rightarrow> (c\<^sub>2, Normal s)"
int_pow_eq: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and m :: "int"   and n :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> n) = (int (group.ord G x) dvd n - m)"
function_ring_car_closed: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier (function_ring (carrier R) R)" shows "f a \<in> carrier R"
cong_add: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d"   and d :: "'d"   and e :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (plus b d) (plus c e) a"
ide_eval_can: fixes f :: "'c local.term set" assumes "\<F>C.can f" shows "D.ide (E.map f)"
assms: shows "fst i = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s) \<noteq> 1 \<and> get_S (cpu_reg_val PSR s) \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s)) + 1) mod NWINDOWS)) (cpu_reg_val WIM s) = 0 \<and> get_addr (snd i) s AND 3 = 0"
cr: shows "count_roots_interval_sf (real_of_int_poly p) = (cr, nr)"
excluded_m: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and x :: "name"   and N :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>"   and "x:N \<in> \<parallel>(B)\<parallel>" shows "(<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>) \<or> x:N \<in> BINDINGn B (\<parallel><B>\<parallel>)) \<or> \<not> (<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>) \<or> x:N \<in> BINDINGn B (\<parallel><B>\<parallel>))"
set_mset_mset: fixes xs :: "'b list" shows "set_mset (mset xs) = set xs"
single_eq_single: fixes a :: "'a"   and b :: "'a" shows "(a, {#} = b, {#}) = (a = b)"
less_imp_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
length_word_rsplit_lt_size: fixes n :: "nat"   and m :: "nat"   and w :: "'d word" assumes "n = LENGTH('c)" shows "(m < length (word_rsplit w)) = (m * n < size w)"
w\<^sub>f_def: shows "w\<^sub>f = f\<^sub>0g\<^sub>1.p\<^sub>1 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1"
distinct_1: fixes a :: "'d"   and b' :: "'e" shows "Inl a \<noteq> Inr b'"
dg_Par_Dom_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Par \<alpha>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> dg_Par \<alpha>\<lparr>Obj\<rparr>"
ap_envir_apply: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'f"   and i :: "'e" shows "(f \<diamondop> x) i = f i (x i)"
monotonicity: fixes bullet :: "'b \<Rightarrow> 'b"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "z_property bullet R"   and "(a, b) \<in> R\<^sup>*" shows "(bullet a, bullet b) \<in> R\<^sup>*"
le_neq_trans: fixes a :: "'c list"   and b :: "'c list" assumes "prefix a b"   and "a \<noteq> b" shows "strict_prefix a b"
insertList_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar q" shows "SkewBinomialHeapStruc.invar (SkewBinomialHeapStruc.insertList ts q)"
wgraph_by_map_axioms: shows "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"
prems_2: fixes Ks :: "'a list" assumes "set Ks \<subseteq> K"   and "combine Ks (u # Us) = \<zero>" shows "set (take (length (u # Us)) Ks) \<subseteq> {\<zero>}"
Minkowski_inequality_le_1_2: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "0 < p"   and "p \<le> 1"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"   and "integrable M (\<lambda>x. \<bar>g x\<bar> powr p)" shows "(LINT x|M. \<bar>f x + g x\<bar> powr p) powr (1 / p) \<le> 2 powr (1 / p - 1) * (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p) + 2 powr (1 / p - 1) * (LINT x|M. \<bar>g x\<bar> powr p) powr (1 / p)"
subst_atms_def: fixes AA :: "'a set"   and \<sigma> :: "'s" shows "AA \<cdot>as \<sigma> = (\<lambda>A. A \<cdot>a \<sigma>) ` AA"
match_def: fixes \<mu> :: "('m \<times> 'w) set" shows "match \<mu> = (inj_on fst \<mu> \<and> inj_on snd \<mu> \<and> \<mu> \<subseteq> (\<Union>m. {m} \<times> Field (Pm m)) \<inter> (\<Union>w. Field (Pw w) \<times> {w}))"
elims_3: fixes x :: "('fsym, 'tp) efsym trm" assumes "\<not> TE.wt x" obtains f :: "('fsym, 'tp) efsym" and Tl :: "('fsym, 'tp) efsym trm list" where "x = Fn f Tl" and "\<not> (TE_wtFsym f \<and> list_all TE.wt Tl \<and> TE_arOf f = map TE.tpOf Tl)"
star_aci_3: fixes a :: "assn"   and b :: "assn"   and c :: "assn" shows "a * (b * c) = b * (a * c)"
vfinite_vunion: fixes A :: "V"   and B :: "V" shows "vfinite (A \<union>\<^sub>\<circ> B) = (vfinite A \<and> vfinite B)"
connectedI: fixes "open" :: "'a set \<Rightarrow> bool"   and U :: "'a set" assumes "class.topological_space open"   and "\<And>A B. \<lbrakk>open A; open B; A \<inter> U \<noteq> {}; B \<inter> U \<noteq> {}; A \<inter> B \<inter> U = {}; U \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> False" shows "topological_space.connected open U"
R_O_Id: fixes R :: "('b \<times> 'c) set" shows "R O Id = R"
sg: shows "subgraph G\<^sub>1 G\<^sub>2"
winning_path_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and P :: "'c llist" assumes "ParityGame G" shows "ParityGame.winning_path G p P \<equiv> \<not> lfinite P \<and> (\<exists>a\<in>ParityGame.path_inf_priorities G P. (\<forall>b\<in>ParityGame.path_inf_priorities G P. a \<le> b) \<and> (if p = Even then even else odd) a) \<or> \<not> lnull P \<and> lfinite P \<and> llast P \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<or> lnull P \<and> p = Even"
supp_Abs_1: fixes as :: "atom set"   and x :: "'b" shows "supp ([as]set. x) = supp x - as"
not_alw_iff: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" shows "(\<not> alw P \<omega>) = ev (\<lambda>xs. \<not> P xs) \<omega>"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
Suc_4: shows "M1 \<in> carrier_mat m m"
sg_imp_mix_sg: fixes x :: "'a pmf"   and y :: "'a pmf"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and outcomes :: "'a set"   and \<alpha> :: "real" assumes "x \<succeq>[\<R>] y \<and> (y, x) \<notin> \<R>"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "\<alpha> \<in> {0<..1}" shows "mix_pmf \<alpha> x y \<succeq>[\<R>] y \<and> (y, mix_pmf \<alpha> x y) \<notin> \<R>"
b: shows "dom\<^bsub>CatCod F\<^esub> (F ## g) = F @@ dom\<^bsub>CatDom F\<^esub> g"
summable_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "summable f" shows "summable (\<lambda>n. f (n + k))"
a_inv_def: fixes R :: "('c, 'd) ring_scheme" shows "a_inv R = m_inv (add_monoid R)"
le_iff_add: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
principal_dchar_not_coprime: fixes k :: "nat"   and n :: "nat" assumes "\<not> coprime k n" shows "\<chi>\<^sub>0\<^bsub>n\<^esub> k = 0"