hcju/isabelleps huggingface dataset:
path_inf_priorities_is_nonempty: fixes P :: "'a llist" assumes "valid_path P"   and "\<not> lfinite P" shows "\<exists>k. k \<in> path_inf_priorities P"
is_hpair: fixes x :: "hf"   and y :: "hf" shows "is_hpair \<langle>x, y\<rangle>"
constr: fixes name :: "name"   and C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and ts :: "sterm list"   and us :: "value list" assumes "name |\<in>| constructors.C C_info"   and "list_all2 (special_constants.veval C_info rs \<Gamma>) ts us" shows "special_constants.veval C_info rs \<Gamma> (name $$ ts) (Vconstr name us)"
fmsubset_pred: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "fmpred P m"   and "n \<subseteq>\<^sub>f m" shows "fmpred P n"
read_only_share_all_until_volatile_write_unacquired': fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "ownership_distinct ts"   and "read_only_unowned \<S> ts"   and "weak_sharing_consis ts"   and "\<forall>i<length ts. let (uu_, uu_, uu_, sb, uu_, \<O>, \<R>) = ts ! i in a \<notin> acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O> \<and> a \<notin> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"   and "a \<in> read_only \<S>" shows "a \<in> read_only (share_all_until_volatile_write ts \<S>)"
open_greaterThan: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThan less a)"
a2: fixes dl :: "'a dlist"   and x :: "'a" assumes "\<not> member dl x"   and "P dl" shows "P (DList.insert x dl)"
Arg_eq: fixes z :: "complex" assumes "z \<noteq> 0" shows "z = complex_of_real (cmod z) * exp (\<i> * complex_of_real (Arg z))"
transfer: shows "pcr_\<Pi>\<^sub>1 (\<lambda>u s w. case u of \<omega>\<upsilon> x \<Rightarrow> ConcreteInWorld x w | \<sigma>\<upsilon> \<sigma> \<Rightarrow> False) E!"
observable2_equiv_observable: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set" shows "observable2 ob P = observable ob P"
ordertype_VWF_inc_eq: fixes A :: "V set"   and \<pi> :: "V \<Rightarrow> V" assumes "A \<subseteq> ON"   and "\<pi> ` A \<subseteq> ON"   and "small A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x < y\<rbrakk> \<Longrightarrow> \<pi> x < \<pi> y" shows "ordertype (\<pi> ` A) VWF = ordertype A VWF"
tminus_cancel_1: fixes a :: "real" assumes "0 \<le> a" shows "a + 1 \<ominus> 1 = a"
sup_left_idem: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
prob_space_P: fixes J :: "nat set" assumes "finite J"   and "J \<subseteq> UNIV" shows "prob_space (CI J)"
decomp_vars_1: fixes t :: "('fun, 'var) Term.term" shows "wfrestrictedvars\<^sub>s\<^sub>t (decomp t) = fv t"
sorted_wrt_drop: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (drop n xs)"
spr_simAction: 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 "set (local.spr_simAction a ec) = set (jAction SPR.MC t a)"
rsp: shows "fractrel (0::'b, 1::'b) (0::'b, 1::'b)"
meval_sched_app: fixes as :: "nat list"   and gc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and gc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and bs :: "nat list"   and gc'' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.meval_sched_abv gc as gc'"   and "abs.meval_sched_abv gc' bs gc''" shows "abs.meval_sched_abv gc (as @ bs) gc''"
carrier_update_3: fixes S :: "'a set" shows "\<one>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub> = \<one>"
permute_pure: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm"   and x :: "'b" assumes "class.pure permute" shows "permute p x = x"
S: shows "(\<lambda>c. if c \<in> n then {make_lit (\<not> v) a} \<union> c else c) ` restrict_cnf_atom a v S \<subseteq> S"
normalize_append_cancel_to: fixes l1 :: "(bool \<times> 'd) list"   and l1' :: "(bool \<times> 'd) list"   and l2 :: "(bool \<times> 'd) list"   and l2' :: "(bool \<times> 'd) list" assumes "cancels_to l1 l1'"   and "cancels_to l2 l2'" shows "Cancelation.normalize (l1 @ l2) = Cancelation.normalize (l1' @ l2')"
t_stop_nonneg: fixes a :: "real"   and v :: "real" assumes "braking_movement a v" shows "0 \<le> movement.t_stop a v"
distinct_5: fixes x2 :: "'b"   and x3 :: "bool" shows "IMor x2 \<noteq> IBool x3"
step_deterministic: fixes F_empty :: "'a"   and F_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and F_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and F_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and heap_empty :: "'f"   and heap_get :: "'f \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'f \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'f"   and heap_to_list :: "'f \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and s1 :: "('a, 'f, ('b, 'c) frame) state"   and s2 :: "('a, 'f, ('b, 'c) frame) state"   and s3 :: "('a, 'f, ('b, 'c) frame) state" assumes "std F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>"   and "std.step F_get heap_get heap_add is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> s1 s2"   and "std.step F_get heap_get heap_add is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> s1 s3" shows "s2 = s3"
split_induct: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and P :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'x set \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) set \<Rightarrow> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<Rightarrow> bool"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "\<And>t U S. \<lbrakk>U \<subseteq> X; finite S; 0 \<in> S\<rbrakk> \<Longrightarrow> P t U S ([(monomial (1::'b) t, U)], [])"   and "\<And>t U S. \<lbrakk>U \<subseteq> X; finite S; 0 \<notin> S; S \<inter> .[U] = {}\<rbrakk> \<Longrightarrow> P t U S ([], [(monomial (1::'b) t, U)])"   and "\<And>t U S V x ps0 ps1 qs0 qs1. \<lbrakk>U \<subseteq> X; finite S; 0 \<notin> S; S \<inter> .[U] \<noteq> {}; V \<subseteq> U; S \<inter> .[V] = {}; \<And>V'. \<lbrakk>V' \<subseteq> U; S \<inter> .[V'] = {}\<rbrakk> \<Longrightarrow> card V' \<le> card V; x \<in> U; x \<notin> V; V = max_subset U (\<lambda>V'. S \<inter> .[V'] = {}); x = (SOME x'. x' \<in> U - V); (ps0, qs0) = split t (U - {x}) S; (ps1, qs1) = split (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S); split t U S = (ps0 @ ps1, qs0 @ qs1); P t (U - {x}) S (ps0, qs0); P (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S) (ps1, qs1)\<rbrakk> \<Longrightarrow> P t U S (ps0 @ ps1, qs0 @ qs1)" shows "P t U S (split t U S)"
index_mult_mat_2: fixes A :: "'a Matrix.mat"   and B :: "'a Matrix.mat" shows "dim_row (A * B) = dim_row A"
scale_zero_left: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "Modules.module scale" shows "scale (0::'a) x = (0::'b)"
annotateWhile: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and gs :: "('f \<times> 'e set) list"   and b :: "'e set"   and I :: "'e set"   and V :: "('e \<times> 'e) set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnnoG gs b I V c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P while gs b c Q,A"
simps_4: fixes x2 :: "nonce_t"   and x1 :: "nonce_t" shows "eprivK x2 \<noteq> epublK x1"
inf_left_commute: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
char_poly_matrix_upper_triangular: shows "upper_triangular (char_poly_matrix A)"
set_lebesgue_integral_eq_integral_1: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S" shows "f integrable_on S"
src_dom: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "src\<^sub>D (D.dom \<mu>) = src\<^sub>D \<mu>"
alive_trivial_simps_2: fixes i :: "int"   and s :: "Store" shows "alive (intgV i) s"
exec_gpv_invariant': fixes s :: "'s"   and gpv :: "('d, 'a, 'b) gpv" assumes "I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "set_spmf (exec_gpv callee gpv s) \<subseteq> {(x, s'). I s'}"
coeff_list: fixes p :: "'a list list" shows "map (UP.coeff p) (rev [0..<length p]) = p"
Var: fixes rs :: "dB list"   and rs' :: "dB list"   and x :: "nat" assumes "rs [\<rightarrow>\<^sub>s] rs'" shows "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>s Var x \<degree>\<degree> rs'"
add_neg_numeral_special_7: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus one) = zero"
STSteps_steps_1: fixes S :: "'a"   and P' :: "('a, 'b) Proc"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus" shows "((\<exists>Sa S'. SourceTerm S = SourceTerm Sa \<and> P' = SourceTerm S' \<and> Sa \<longmapsto>Source* S') \<or> (\<exists>T T'. SourceTerm S = TargetTerm T \<and> P' = TargetTerm T' \<and> T \<longmapsto>Target* T')) = (\<exists>S'. P' = SourceTerm S' \<and> S \<longmapsto>Source* S')"
isomorphic_reflexive: fixes f :: "'a \<times> 'a" assumes "VV.ide f" shows "VV.isomorphic f f"
simps: fixes T :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b list" assumes "enumerateable T less" shows "enumerateable.sorted less a = (a = [] \<or> (\<exists>xs x. a = x # xs \<and> (\<forall>y\<in>set xs. less x y) \<and> enumerateable.sorted less xs))"
primitive_root_unity_def: fixes k :: "nat"   and x :: "'a" shows "primitive_root_unity k x = (k \<noteq> 0 \<and> x ^ k = (1::'a) \<and> (\<forall>k'<k. k' \<noteq> 0 \<longrightarrow> x ^ k' \<noteq> (1::'a)))"
Pow_insert: fixes a :: "'b"   and A :: "'b set" shows "Pow (insert a A) = Pow A \<union> insert a ` Pow A"
assms_3: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "(Q, P) \<in> X"
s_2: fixes i :: "nat" shows "s i \<in> {a..}"
find_first_later: fixes i :: "nat"   and xs :: "'a list"   and j :: "nat"   and x :: "'a" assumes "i < length xs"   and "j < length xs"   and "i < j"   and "xs ! i = x"   and "xs ! j = x" shows "find_first x xs \<noteq> Some j"
add_pos_pos: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "{#} \<subset># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
HEndPhase0_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p" shows "allBlocks s' \<subseteq> allBlocks s \<union> {dblock s' p}"
sinvar_mono_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool" shows "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar = (\<forall>nP N E' E. wf_graph \<lparr>nodes = N, edges = E\<rparr> \<and> E' \<subseteq> E \<and> sinvar \<lparr>nodes = N, edges = E\<rparr> nP \<longrightarrow> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP)"
b: shows "b = coeff (pCons a r) (degree y) / lead_coeff y"
subdiv_ends_G_rev_1: shows "tail G (rev_G uv) = v"
0: fixes i :: "nat" shows "Cring_Poly.compose R (g i) q = p i \<odot>\<^bsub>P\<^esub> q [^]\<^bsub>P\<^esub> i"
perp_in_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X C D A B"
mat_mult_sub_lattice: fixes fs :: "'a vec list"   and n :: "nat"   and gs :: "'a vec list"   and A :: "int mat" assumes "set fs \<subseteq> carrier_vec n"   and "set gs \<subseteq> carrier_vec n"   and "A \<in> carrier_mat (length fs) (length gs)"   and "mat_of_rows n fs = of_int_hom.mat_hom A * mat_of_rows n gs" shows "vec_module.lattice_of n fs \<subseteq> vec_module.lattice_of n gs"
ord_le_eq_trans: fixes a :: "('c, 'd) Term.term"   and b :: "('c, 'd) Term.term"   and c :: "('c, 'd) Term.term" assumes "a \<sqsubseteq> b"   and "b = c" shows "a \<sqsubseteq> c"
br_termrel_def: fixes \<delta> :: "('d, 'f) ta_rule set" shows "br_termrel \<delta> \<equiv> {(Q', Q). Q \<subset> Q' \<and> Q' \<subseteq> b_accessible \<delta>} <*lex*> finite_psubset"
f: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "Knuth_ex_4_6_2_22_main.w' q_ty a b u v w * #v + Knuth_ex_4_6_2_22_main.v' q_ty b u v w * #w = Knuth_ex_4_6_2_22_main.f q_ty u v w"
induct: fixes P :: "nat list \<Rightarrow> nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<Rightarrow> (('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "nat"   and a2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a3 :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec" assumes "\<And>i bezout Pa A Q. P [] i bezout (Pa, A, Q)"   and "\<And>j xs i bezout Pa A Q. \<lbrakk>A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j \<Longrightarrow> P xs i bezout (Pa, A, Q); \<And>x xa y xb ya xc yb xd yc xe xf xg yd. \<lbrakk>\<not> A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j; x = bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j); (xa, y) = x; (xb, ya) = y; (xc, yb) = ya; (xd, yc) = yb; xe = diagonal_step A i j yc xd; xf = diagonal_step_PQ A i j bezout; (xg, yd) = xf\<rbrakk> \<Longrightarrow> P xs i bezout (xg ** Pa, xe, Q ** yd)\<rbrakk> \<Longrightarrow> P (j # xs) i bezout (Pa, A, Q)" shows "P a0 a1 a2 a3"
assms_1: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
aux_comp_with_cmod: fixes n :: "nat"   and g :: "nat \<Rightarrow> nat" assumes "(\<forall>x<2 ^ n. g x = 0) \<or> (\<forall>x<2 ^ n. g x = 1)" shows "(cmod (\<Sum>k<2 ^ n. (- 1) ^ g k))\<^sup>2 = 2 ^ (2 * n)"
seq': fixes rs :: "'c rule list"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t'"
closure_of_singleton: fixes X :: "'a topology"   and a :: "'a" assumes "t1_space X" shows "X closure_of {a} = (if a \<in> topspace X then {a} else {})"
hyps_3: shows "SubstForm v (q_Eats v w) x xevw"
pr_naturality_2: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "\<pp>\<^sub>1[c, d] \<cdot> (f \<otimes> g) = f \<cdot> \<pp>\<^sub>1[a, b]"
app_def: fixes f :: "ZF"   and x :: "ZF" shows "f \<acute> x \<equiv> THE y. Elem (Opair x y) f"
length_1_calculate_data_satisfies_properties_R: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "length qs = 1" shows "satisfies_properties_R p qs (get_subsets_R (calculate_data_R p qs)) (get_signs_R (calculate_data_R p qs)) (get_matrix_R (calculate_data_R p qs))"
NMPcharn: fixes a :: "'a"   and p :: "'a list"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" assumes "a \<in> set p"   and "dom (C a) \<noteq> {}" shows "not_MT C p"
derived_set_of_restrict: fixes X :: "'c topology"   and S :: "'c set" shows "X derived_set_of (topspace X \<inter> S) = X derived_set_of S"
iall_conj_distrib: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<box> t I. P t \<and> Q t) = ((\<box> t I. P t) \<and> (\<box> t I. Q t))"
length_drop: fixes n :: "nat"   and xs :: "'k list" shows "length (drop n xs) = length xs - n"
bounded_iff: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a \<le> b \<sqinter> c) = (a \<le> b \<and> a \<le> c)"
partial_order_onD_2: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "partial_order_on A r" shows "trans r"
cindex_poly_inverse_add: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "coprime p q" shows "cindex_poly a b q p + cindex_poly a b p q = cindex_poly a b 1 (q * p)"
d_IN_mono: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. f (x, y) \<le> g (x, y)" shows "d_IN f y \<le> d_IN g y"
RP_step_imp_GC_step: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and St' :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" assumes "St \<leadsto>RP St'" shows "lclss_of_state St \<leadsto>GC lclss_of_state St'"
traces_failures: fixes xs :: "'c list"   and P :: "'c process" assumes "xs \<in> traces P" shows "(xs, {}) \<in> failures P"
of_weak_ranking_weak_ranking_aux': fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "A \<subseteq> carrier"   and "x \<in> A"   and "y \<in> A" shows "of_weak_ranking (weak_ranking_aux A) x y = restrict_relation A le x y"
bexI: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and A :: "'c set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
eval_pCons: fixes x :: "'a" shows "poly (pCons (0::'a) 1) x = x"
NOU: shows "Oadmit \<sigma> ODE (BVO ODE)"
finite_V_Fe: fixes G :: "('b, 'c) gb_graph_rec_scheme"   and A :: "'b set" assumes "gb_graph G"   and "finite (g_V G)"   and "A \<in> gbg_F G" shows "finite A"
inject_10: fixes x101 :: "nat"   and x102 :: "msg"   and y101 :: "nat"   and y102 :: "msg" shows "(Crypt x101 x102 = Crypt y101 y102) = (x101 = y101 \<and> x102 = y102)"
base_out: fixes d :: "nat"   and dm :: "nat"   and ds :: "nat set"   and p :: "(nat \<times> int) list" assumes "d < dm"   and "d \<notin> ds"   and "p \<in> sparsegrid' dm" shows "base ds p ! d = p ! d"
rel_FGl_pos_distr_cond_def: fixes Co1 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Co1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co3 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co3' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co4 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co4' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra4 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra4' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and uu :: "('ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao \<times> 'ap \<times> 'aq \<times> 'ar \<times> 'as \<times> 'at \<times> 'au) itself" shows "rel_FGl_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' uu = (\<forall>L1 L1' L2 L2' L3 L3'. rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 OO rel_FGl L1' L2' L3' Co1' Co2' Co3' Co4' Contra1' Contra2' Contra3' Contra4' \<le> rel_FGl (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4'))"
1: shows "real (card G) - real (card (pseudoAllocation aa \<inter> pseudoAllocation a)) = real (sum (pseudoAllocation a <| (N \<times> finestpart G)) (pseudoAllocation a)) - real (sum (pseudoAllocation a <| (N \<times> finestpart G)) (pseudoAllocation aa))"
power2_i: shows "\<i>\<^sup>2 = - 1"
termi: shows "\<Gamma>\<turnstile>c \<down> s"
closedin_diff: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "closedin U S"   and "openin U T" shows "closedin U (S - T)"
Red_F_\<G>_def: fixes N :: "'f set" shows "empt_ord.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F C. D \<in> Red_F_G (\<G>_Fset N) \<or> (\<exists>E\<in>N. False \<and> D \<in> \<G>_F E)}"
crsp_step_dec_b_e: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and n :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and e :: "nat" assumes "ly = layout_of ap"   and "inv_locate_a (as, lm) (n, l, r) ires"   and "abc_lm_v lm n = 0"   and "abc_fetch as ap = Some (Dec n e)" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Dec n e))) (steps (start_of ly as + 2 * n, l, r) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp) ires"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "l2_state.bad \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, chan = chan, bad = bad, \<dots> = more\<rparr> = bad"
map_def: fixes f :: "'a2 \<times> 'a1" shows "S.map f = (if A2xA1.arr f then (snd f, fst f) else A1xA2.null)"
IH: shows "e_nth (local.prefixes j t) x = e_nth (local.prefixes j (t + d)) x"
floor_mono: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
1: shows "[\<^bold>\<not>\<^bold>\<box>(\<^bold>\<not>\<phi> \<^bold>\<or> \<^bold>\<box>\<^bold>\<not>\<psi>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<box>\<^bold>\<not>\<phi> \<^bold>\<or> \<^bold>\<box>\<^bold>\<not>\<psi>) in v]"
list_all_eq_const_imp_replicate: fixes y :: "'a"   and xs :: "'a list" assumes "list_all (\<lambda>x. x = y) xs" shows "xs = replicate (length xs) y"
simps: fixes a :: "'b llist"   and A :: "'b set" shows "(a \<in> A\<^sup>\<star>) = (a = LNil \<or> (\<exists>l aa. a = aa ## l \<and> l \<in> A\<^sup>\<star> \<and> aa \<in> A))"
sel_1: fixes x1 :: "bool"   and x2 :: "'b \<Rightarrow> 'b language" shows "\<oo> (Lang x1 x2) = x1"