hcju/isabelleps huggingface dataset:
ENR_def: fixes S :: "'c set" shows "ENR S \<equiv> \<exists>U. open U \<and> S retract_of U"
transp_gr: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp (\<lambda>x y. less y x)"
bijective_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.bijective f = (\<forall>j. \<exists>i. f (i, j) = top \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
g: shows "g = (\<lambda>t. \<Sum>i\<in>K. z2 i * \<chi> (C i) t)"
ground_subst_exists: fixes V :: "'b set" assumes "finite V" shows "\<exists>\<sigma>. ground_on V \<sigma>"
add_left_mono: fixes a :: "complex"   and b :: "complex"   and c :: "complex" assumes "a \<le>\<^sub>\<complex> b" shows "c + a \<le>\<^sub>\<complex> c + b"
f12: shows "\<forall>x0. (0 < x0) = (\<not> x0 \<le> 0)"
bdd_aboveI2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'b" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "preorder.bdd_above less_eq (f ` A)"
env_delete_noop: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e" assumes "x \<notin> edom m" shows "env_delete x m = m"
inject_2: fixes x7 :: "char list"   and y7 :: "char list" shows "(Goto x7 = Goto y7) = (x7 = y7)"
whileLoop_def: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) set \<times> bool" shows "whileLoop C   B \<equiv> \<lambda>r s. ({(r', s'). (Some (r, s), Some (r', s')) \<in> whileLoop_results C B}, (Some (r, s), None) \<in> whileLoop_results C B \<or> \<not> whileLoop_terminates C B r s)"
SourceExistsDSource: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "S \<in> Sources i C \<longrightarrow> (\<exists>Z. S \<in> DSources i Z)"
lossless: fixes x :: "'a"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "lossless_spmf (callee s x)"
zip_same_conv_map: fixes xs :: "'b list" shows "zip xs xs = map (\<lambda>x. (x, x)) xs"
abstract_LP_def: fixes A :: "'a mat"   and b :: "'a vec"   and c :: "'a vec"   and m :: "nat"   and n :: "nat" shows "abstract_LP A b c m n \<equiv> b \<in> carrier_vec m \<and> c \<in> carrier_vec n \<and> A \<in> carrier_mat m n"
U_5: shows "rec_inseparable {uu_. \<exists>x vs'. uu_ = x \<and> ((2, \<lambda>n. if n = 0 then x else 0), 0, vs') \<in> (step U)\<^sup>*} {uu_. \<exists>x vs'. uu_ = x \<and> ((2, \<lambda>n. if n = 0 then x else 0), 1, vs') \<in> (step U)\<^sup>*}"
admits_SNF: shows "\<forall>A. square_mat A \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
keys_is_none_rep: fixes k :: "'b"   and m :: "('b, 'c) mapping" shows "(k \<in> Mapping.keys m) = (\<not> Option.is_none (Mapping.lookup m k))"
GSubmodule_GSpan_single: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and m :: "'c" assumes "FinGroupRepresentation G smult V"   and "m \<in> V" shows "RModule (aezfun_setspan G) smult (R_scalar_mult.RSpan (aezfun_setspan G) smult [m]) \<and> R_scalar_mult.RSpan (aezfun_setspan G) smult [m] \<subseteq> V"
super_orientable_def: fixes x :: "'a" shows "bounded_distrib_allegory.super_orientable (\<sqinter>) (\<le>) (\<squnion>) bot id x \<equiv> \<exists>y. x \<le> y \<squnion> id y \<and> y \<sqinter> id y = bot"
continuous_map_componentwise_Euclidean_space: fixes X :: "'a topology"   and n :: "nat"   and f :: "'a \<Rightarrow> nat \<Rightarrow> real" shows "continuous_map X (Euclidean_space n) (\<lambda>x i. if i < n then f x i else 0) = (\<forall>i<n. continuous_map X euclideanreal (\<lambda>x. f x i))"
ereal_uminus_le_0_iff: fixes a :: "ereal" shows "(- a \<le> 0) = (0 \<le> a)"
rank_def: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and v :: "nat \<times> 'd" shows "rank A w v \<equiv> GREATEST k. v \<in> graph A w k"
simps: fixes "consider" :: "'e \<Rightarrow> bool"   and gpv :: "('f, 'e, 'g) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
finite_measure_subadditive_finite: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and p :: "'b spmf" assumes "finite I"   and "A ` I \<subseteq> sets (measure_spmf p)" shows "Sigma_Algebra.measure (measure_spmf p) (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure (measure_spmf p) (A i))"
rev_curr_def: fixes f :: "'d \<times> 'c \<Rightarrow> 'e"   and b :: "'c" shows "local.rev_curr f b = (if b \<in> Field t then \<lambda>a. f (a, b) else undefined)"
eq_height_0: fixes t :: "'c tree" shows "(height t = 0) = (t = \<langle>\<rangle>)"
small_vintersection_set: fixes A :: "V"   and B :: "V" shows "small {x. x \<in>\<^sub>\<circ> A \<and> x \<in>\<^sub>\<circ> B}"
aassoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A" shows "a \<plusminus> b \<plusminus> c = a \<plusminus> (b \<plusminus> c)"
nFOR_disjuncts: fixes \<psi> :: "('a, 'b) aformula" assumes "nf_ACI \<psi>" shows "nFOR (disjuncts_list \<psi>) = \<psi>"
subst_minus: fixes L :: "'a literal"   and \<mu> :: "'s" shows "- L \<cdot>l \<mu> = - (L \<cdot>l \<mu>)"
cols_take_cols_subset: fixes A :: "'b mat"   and inds :: "nat list" shows "set (cols (take_cols A inds)) \<subseteq> set (cols A)"
acc_modi_Package_le_cases: fixes m :: "acc_modi" assumes "Package \<le> m" obtains "m = Package"   | "m = Protected"   | "m = Public"
extend_chain: fixes A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c"   and n :: "nat" shows "extend A S f n \<subseteq> extend A S f (Suc n)"
iSPEC_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'c \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "SPEC \<Phi> \<le> SPEC \<Psi>"
term_ok_imp_typ_ok': 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"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t" shows "typ_ok \<Theta> (the (typ_of t))"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * (a * c) = a * (b * c)"
xs: shows "(0::'a) \<notin> snd ` set xs"
rel_spmf_restrict_relpI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf"   and P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "rel_spmf R p q"   and "pred_spmf P p"   and "pred_spmf Q q" shows "rel_spmf (R \<upharpoonleft> P \<otimes> Q) p q"
divide_formula: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "y \<noteq> \<zero>"   and "val_Zp y \<le> val_Zp x" shows "y \<otimes> local.divide x y = x"
reducing_sys_satisfies_properties: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list"   and signs :: "rat list list"   and matrix :: "rat mat" assumes "p \<noteq> 0"   and "0 < length qs"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "satisfies_properties p qs subsets signs matrix" shows "satisfies_properties p qs (get_subsets (reduce_system p (qs, matrix, subsets, signs))) (get_signs (reduce_system p (qs, matrix, subsets, signs))) (get_matrix (reduce_system p (qs, matrix, subsets, signs)))"
abs_conv_abscissa_MInfI: fixes f :: "'a fds" assumes "\<And>s. fds_abs_converges f s" shows "abs_conv_abscissa f = - \<infinity>"
Seq_def: fixes f :: "unit \<Rightarrow> 'b Predicate.seq" shows "Predicate.Seq f = Predicate.pred_of_seq (f ())"
absorb_iff1: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subseteq># a) = (a \<union># b = a)"
candidate_free_theorem: fixes h :: "'c \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zs :: "'c list" assumes "h (f x y) = g (h x) (h y)" shows "map h (candidate f zs) = candidate g (map h zs)"
euler_trail_conv_connected: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "Digraph_Component.connected G" shows "euler_trail u p v = (trail u p v \<and> set p = arcs G)"
homogeneousI: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>s t. \<lbrakk>s \<in> keys p; t \<in> keys p\<rbrakk> \<Longrightarrow> deg_pm s = deg_pm t" shows "homogeneous p"
prime_power_exp_nat: fixes p :: "nat"   and n :: "nat"   and x :: "nat"   and k :: "nat" assumes "prime p"   and "n \<noteq> 0"   and "x ^ n = p ^ k" shows "\<exists>i. x = p ^ i"
SD_singleton_left: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and q :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier"   and "q \<in> lotteries_on carrier" shows "SD le (return_pmf x) q = (\<forall>y\<in>set_pmf q. le x y)"
select1: shows "primitive_extractor (is_Iiface, iiface_sel) m = (i_matches, rest1)"
labels_rel': shows "labels_rel = (\<lambda>gq gp. (gq, gp) \<in> {((\<Gamma>, q), \<Gamma>', p). \<Gamma> = \<Gamma>' \<and> p \<leadsto>\<^bsub>\<Gamma>\<^esub> q})"
is_terminal_def: fixes s :: "symbol" shows "is_terminal s = (s \<in> \<TT>)"
assms: shows "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"
fps_binomial_minus_of_nat: fixes n :: "nat" shows "fps_binomial (- of_nat n) = inverse ((1 + fps_X) ^ n)"
ball_subseteq: fixes s1 :: "'e set"   and s2 :: "'e set" shows "(\<forall>x\<in>s1. x \<in> s2) = (s1 \<subseteq> s2)"
mudiff: shows "gs1.\<mu> i j - rat_of_int c = gs2.\<mu> i j"
space_F: fixes \<Omega> :: "'c set"   and F :: "'d \<Rightarrow> 'c measure"   and i :: "'d" assumes "filtration \<Omega> F" shows "space (F i) = \<Omega>"
inf_same_context: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y \<sqinter> z"   and "y \<le> x \<sqinter> z" shows "x \<sqinter> z = y \<sqinter> z"
Diff_partition: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
sset_flat: fixes s :: "'d list stream" assumes "\<forall>xs\<in>sset s. xs \<noteq> []" shows "sset (flat s) = \<Union> (set ` sset s)"
hyps: fixes \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "a \<in> all_shared sb \<union> all_unshared sb" shows "share sb \<S>' a = share sb \<S> a"
equiv_\<phi>: fixes x :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "pred_stream (\<lambda>s. s \<in> S) x" shows "pred_stream (\<lambda>s. \<phi> (reps (abss s)) = \<phi> s) (state cfg' ## x)"
fds_nth_subseries: fixes P :: "nat \<Rightarrow> bool"   and f :: "'a fds"   and n :: "nat" shows "fds_nth (fds_subseries P f) n = (if P n then fds_nth f n else (0::'a))"
not_distinct_decomp_min_not_distinct: fixes ws :: "'c list" assumes "\<not> distinct ws" shows "\<exists>xs y ys zs. ws = xs @ y # ys @ y # zs \<and> distinct (ys @ [y])"
set_plus_def: fixes A :: "'c set"   and B :: "'c set" shows "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
id: shows "complex_of_int = complex_of_real \<circ> real_of_int"
pax4: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "|~ \<box>[P]_v = (_actrans P v \<and> \<circle>\<box>[P]_v)"
2: shows "v \<notin> f ` set (remdups_wrt_rev f xs (f x # vs))"
redT_updTs_preserves: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and ts :: "'b \<Rightarrow> ('a \<times> 'g \<Rightarrow>f nat) option"   and m :: "'c"   and t :: "'b"   and x :: "'a"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and x' :: "'a"   and m' :: "'c"   and ln :: "'g \<Rightarrow>f nat"   and ln' :: "'g \<Rightarrow>f nat" assumes "lifting_wf final r P"   and "ts_ok P ts m"   and "r t (x, m) ta (x', m')"   and "ts t = \<lfloor>(x, ln)\<rfloor>"   and "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "ts_ok P (redT_updTs ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', ln'))) m'"
Entry_target: fixes a :: "'edge" assumes "valid_edge a"   and "targetnode a = (_Entry_)" shows "False"
vifunionI: fixes i :: "V"   and I :: "V"   and a :: "V"   and f :: "V \<Rightarrow> V" assumes "i \<in>\<^sub>\<circ> I"   and "a \<in>\<^sub>\<circ> f i" shows "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
eqI: fixes x :: "'a"   and y :: "'a" assumes "proj x = proj y"   and "norm x = norm y" shows "x = y"
cos_zero_iff_int: fixes x :: "real" shows "(cos x = 0) = (\<exists>i. odd i \<and> x = real_of_int i * (pi / 2))"
sqrt_sqrt: fixes x :: "real" assumes "0 \<le> x" shows "sqrt x * sqrt x = x"
mono_rules_2: fixes P :: "'b process"   and Q :: "'b process"   and c :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "P \<sqsubseteq> Q" shows "(c`!`x \<rightarrow> P) \<sqsubseteq> (c`!`x \<rightarrow> Q)"
of_rat_inverse: fixes a :: "rat" shows "of_rat (inverse a) = inverse (of_rat a)"
hom_int_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) f (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) = hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) f x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n"
fin_def: fixes A :: "'d \<times> 'e \<times> 'c" shows "fin A = snd (snd A)"
assms_2: shows "\<forall>p. path_entry E p n' \<longrightarrow> n \<in> set p"
assms_2: fixes si :: "'d"   and s :: "'c" assumes "(si, s) \<in> R" shows "ci si = c s"
intros_1: fixes s :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> reachable A I"
rev_image_eqI: fixes x :: "'a"   and A :: "'a set"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
future_bounded_mfodl_def: fixes \<phi> :: "Formula.formula" shows "future_bounded_mfodl \<phi> \<equiv> Formula.future_bounded \<phi>"
composition_series_length_one: shows "(length \<HH> = 1) = (\<HH> = [{\<one>\<^bsub>G\<^esub>}])"
spr_simObs: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "local.spr_simObs a ec = envObs a (tLast t)"
assms_3: shows "A B C Cong3 A' B' C1"
ucast_eq: fixes w :: "'c word" shows "UCAST('c \<rightarrow> 'b) w = word_of_int (uint w)"
relation_of_Hide: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "action.relation_of (P \ cs) = (R (\<lambda>(S, S'). \<exists>s. diff_tr S' S = tr_filter (s - tr S) cs \<and> action.relation_of P (S, S'\<lparr>tr := s, ref := ref S' \<union> cs\<rparr>)) ;; action.relation_of Skip)"
intros_2: fixes l :: "'d"   and g :: "('e, 'f) cconstraint"   and a :: "'g"   and r :: "'e list"   and l' :: "'d"   and A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and Z :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', (Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'}\<rangle>"
collapse_code_def: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat" shows "collapse_code node_eq_impl node_hash_impl \<equiv> \<lambda>x (xa, xb, xc, xd). idx_of_code node_eq_impl node_hash_impl (xa, xb, xc, xd) x \<bind> (\<lambda>xe. let y' = as_take (xe + 1) xb in RETURN (xa, y', xc, xd))"
Amin_le: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat"   and j :: "nat" assumes "f \<in> {j. j \<le> n} \<rightarrow> Z\<^sub>\<infinity>"   and "j \<in> {k. k \<le> n}" shows "Amin n f \<le> f j"
a_get_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes_locs document_ptr = {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)}"
lt: shows "n (i - 1) - 1 < n (i - 1)"
dupe_monic_i_def: fixes ops :: "'a arith_ops_record"   and D :: "'a list"   and H :: "'a list"   and S :: "'a list"   and T :: "'a list"   and U :: "'a list" shows "dupe_monic_i ops D H S T U = (case pdivmod_monic_i ops (times_poly_i ops T U) D of (Q, R) \<Rightarrow> (plus_poly_i ops (times_poly_i ops S U) (times_poly_i ops H Q), R))"
non_qAFresh_or_in_ordLess_var: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood" shows "|{uu_. \<exists>x X. uu_ = x \<and> (x \<in> V \<or> X \<in> XS \<and> \<not> qAFresh xs x X)}| <o |UNIV|"
collapse: fixes state :: "('a, 'b) state" shows "State (\<T> state) (\<B>\<^sub>i\<^sub>l state) (\<B>\<^sub>i\<^sub>u state) (\<V> state) (\<U> state) (\<U>\<^sub>c state) = state"
q_inter: fixes Q :: "'a set"   and Q' :: "'a set" assumes "i1.proj.quorum Q"   and "i1.proj.quorum Q'"   and "Q \<inter> I\<^sub>1 \<noteq> {}"   and "Q' \<inter> I\<^sub>1 \<noteq> {}" shows "Q \<inter> Q' \<inter> I\<^sub>1 \<noteq> {}"
axioms_1: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b" assumes "dp_consistency_empty lookup update P empty" shows "dp_consistency lookup update P"
intro: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and j :: "real \<times> real"   and i :: "real \<times> real" assumes "typeII_twoCube twoC"   and "valid_two_cube twoC"   and "analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i" shows "green_typeII_cube_axioms i j twoC F"
summable'_2: fixes A :: "nat set" shows "(\<lambda>n. g n / nat_power n s) abs_summable_on A"
simps_2: fixes n :: "nat"   and m :: "nat"   and k :: "nat" shows "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
UniP: fixes H :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "Hintikka H"   and "\<^bold>\<forall> p \<in> H" shows "\<forall>t. p\<langle>t/0\<rangle> \<in> H"
mcont_fun_lub_apply: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'e set \<Rightarrow> 'e"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "mcont luba orda (fun_lub lubb) (fun_ord ordb) f = (\<forall>x. mcont luba orda lubb ordb (\<lambda>y. f y x))"
elims: fixes x :: "Core.term"   and y :: "typ option" assumes "strip_all_single_var x = y" obtains alla :: "String.literal" and S :: "typ" and T :: "typ" and t :: "Core.term" where "x = Ct alla S $ Abs T t" and "y = (if alla = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then Some T else None)"   | v :: "String.literal" and va :: "typ" where "x = Ct v__ va__" and "y = None"   | v :: "variable" and va :: "typ" where "x = Fv vb__ vaa__" and "y = None"   | v :: "nat" where "x = Bv vc__" and "y = None"   | v :: "typ" and va :: "Core.term" where "x = Abs vd__ vab__" and "y = None"   | vb :: "variable" and vc :: "typ" and va :: "Core.term" where "x = Fv vba__ vca__ $ vac__" and "y = None"   | vb :: "nat" and va :: "Core.term" where "x = Bv vbb__ $ vad__" and "y = None"   | vb :: "typ" and vc :: "Core.term" and va :: "Core.term" where "x = Abs vbc__ vcb__ $ vae__" and "y = None"   | vb :: "Core.term" and vc :: "Core.term" and va :: "Core.term" where "x = vbd__ $ vcc__ $ va" and "y = None"   | v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = ve__ $ Ct vbe__ vcd__" and "y = None"   | v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = vf__ $ Fv vbf__ vce__" and "y = None"   | v :: "Core.term" and vb :: "nat" where "x = vg__ $ Bv vbg__" and "y = None"   | v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = v $ (vb $ vc)" and "y = None"
RETURN_SPEC_conv: fixes r :: "'c" shows "RETURN r = SPEC (\<lambda>x. x = r)"
lm135: fixes X :: "'a set"   and f :: "('a \<times> 'b) set" assumes "finite X" shows "MiscTools.sum' f (X \<inter> Domain f) = MiscTools.sum' f X"
trms_irreducible_def: fixes CC :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and S :: "'a eclause set"   and t :: "'a trm" shows "trms_irreducible CC \<sigma> S t = (\<forall>x. (\<exists>x'\<in>trms_ecl CC. occurs_in x (x' \<lhd> \<sigma>)) \<longrightarrow> (x, t) \<in> trm_ord \<longrightarrow> trm_rep x S = x)"
dres_monad1: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a dres" shows "dRETURN x \<bind> f = f x"
pathsConverge': fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and y :: "'node"   and ys :: "'node list"   and z :: "'node" shows "pathsConverge g x xs y ys z = pathsConverge' g x xs y ys z"
basis_enum_of_vec_inverse: fixes v :: "complex vec" assumes "dim_vec v = length canonical_basis" shows "vec_of_basis_enum (basis_enum_of_vec v) = v"
carrier_vec_conjugate: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "conjugate v \<in> carrier_vec n"
finite_convex_hull_iff_convex_hull_list: fixes Vs :: "'a vec set"   and Vsl' :: "'a vec list" assumes "Vs \<subseteq> carrier_vec m"   and "Vs = set Vsl'" shows "m.convex_hull Vs = m.convex_hull_list Vsl'"
sqprojection_def: shows "sqprojection \<equiv> \<lambda>z. z /\<^sub>R infnorm z"
simps_4: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>t := t'\<rangle> # S) = (fv t' \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t) S)"
l_get_parent_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_parent_axioms type_wf known_ptrs get_parent get_parent_locs get_child_nodes"
assms_1: shows "emeasure M (space M) \<le> 1"
less_restrictive_and_finite: fixes atom_ordering :: "('a \<times> 'a) set"   and R1 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and R2 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.less_restrictive R1 R2"   and "propositional_atoms.derived_clauses_are_finite R1" shows "propositional_atoms.derived_clauses_are_finite R2"
assms: shows "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> l"
gamma_differentiable: shows "\<forall>x\<in>{a..b} - s. \<gamma> differentiable at x"