hcju/isabelleps huggingface dataset:
Pairs_def: fixes AAA :: "'a set set" shows "Pairs AAA = concat (linorder.sorted_list_of_set (le_of_comp compare) ((pairs \<circ> linorder.sorted_list_of_set (le_of_comp compare)) ` AAA))"
vcomp_atD: fixes s :: "V"   and r :: "V"   and a :: "V"   and c :: "V" assumes "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = c"   and "vsv r"   and "vsv s"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "\<exists>b. s\<lparr>b\<rparr> = c \<and> r\<lparr>a\<rparr> = b"
min_product_assoc: fixes A :: "'b fset set"   and B :: "'b fset set"   and C :: "'b fset set" shows "(A \<otimes>\<^sub>m B) \<otimes>\<^sub>m C = A \<otimes>\<^sub>m B \<otimes>\<^sub>m C"
completely_multiplicative_function_only_pows: fixes f :: "'a fds"   and p :: "nat" assumes "completely_multiplicative_function (fds_nth f)" shows "completely_multiplicative_function (fds_nth (fds_primepow_subseries p f))"
synth_analz_idem: fixes H :: "msg set" shows "synth (analz (synth (analz H))) = synth (analz H)"
purge_Cons_in: fixes u :: "'U"   and G :: "'U set"   and c :: "'C"   and ucl :: "('U \<times> 'C) list" assumes "u \<notin> G" shows "purge G ((u, c) # ucl) = (u, c) # purge G ucl"
a: fixes b :: "'a set" assumes "b \<in># \<B>"   and "\<forall>x'\<in>{x, y}. x' \<notin> b" shows "x \<notin> b \<and> y \<notin> b"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'd"   and f2 :: "'d"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = ((\<forall>nat. x = enat nat \<longrightarrow> P (f1 nat)) \<and> (x = \<infinity> \<longrightarrow> P f2))"
bounded_linear_axioms_def: fixes f :: "'e \<Rightarrow> 'f" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
bound: shows "\<forall>j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k) \<longrightarrow> j \<le> k"
i_closed: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e" assumes "Group G"   and "a \<in> carrier G" shows "\<rho>\<^bsub>G\<^esub> a \<in> carrier G"
filtercomap_principal: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set" shows "filtercomap f (principal A) = principal (f -` A)"
rejects_state_step: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "1 < s" shows "step drinks s r l i = None"
progressing_appendD2: fixes p :: "('b \<times> 'b) list"   and q :: "('b \<times> 'b) list" assumes "progressing (p @ q)" shows "progressing q"
unit_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>F.unit a : F.map\<^sub>0 a \<Rightarrow>\<^sub>D local.map a\<guillemotright>"
diff_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq b a" shows "less_eq (minus c a) (minus c b)"
distinct_1: fixes x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree" shows "\<langle>\<rangle> \<noteq> \<langle>x21, x22, x23\<rangle>"
simps_3: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int" shows "closest_pair_bf [p\<^sub>0, p\<^sub>1] = (p\<^sub>0, p\<^sub>1)"
subsumes_modulo_def: fixes C :: "('c, 'd) Term.term literal multiset"   and D :: "('c, 'd) Term.term literal multiset"   and \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term option" shows "subsumes_modulo C D \<sigma> = (\<exists>\<tau>. dom \<tau> = vars_clause C \<union> dom \<sigma> \<and> extends_subst \<sigma> \<tau> \<and> subst_cls C (subst_of_map Var \<tau>) \<subseteq># D)"
cake_plus_def: fixes t :: "exp0"   and u :: "exp0" shows "cake_plus t u = App (Opn Plus) [t, u]"
id: shows "[l..<Suc (l + k)] = l # [Suc l..<Suc (Suc l + kk)]"
ODord_def: fixes X :: "'c Order set"   and Y :: "'c Order set" shows "X \<sqsubset> Y = (\<exists>x\<in>X. \<exists>y\<in>Y. \<exists>c\<in>carrier y. ord_equiv x (Iod y (segment y c)))"
forest_components_increasing: fixes x :: "'a" shows "x \<le> top \<sqinter> top"
sp_fundef_generalize: fixes F :: "'c \<Rightarrow> 'd fundef option"   and fd :: "'e fundef"   and xs :: "('dyn, 'f, 'c, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>' :: "Unboxed.type option list" assumes "sp_fundef F fd xs = Ok \<Sigma>'" shows "sp_fundef F fd (map Subx.generalize_instr xs) = Ok (map Map.empty \<Sigma>')"
commute: fixes a :: "'a upred"   and b :: "'a upred" shows "(a \<or> b) = (b \<or> a)"
CardinalityPrognosisLet_axioms: shows "CardinalityPrognosisLet prognosis cHeap Aheap"
induct: fixes P :: "tm \<Rightarrow> bool"   and a0 :: "tm" assumes "\<And>j. P (CP j)"   and "\<And>n. P (tm.Bound n)"   and "\<And>t. P t \<Longrightarrow> P (tm.Neg t)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (tm.Add t s)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (tm.Sub t s)"   and "\<And>i t. (\<And>x. \<lbrakk>x = polynate i; x \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> P t) \<Longrightarrow> P (tm.Mul i t)"   and "\<And>n c t. \<lbrakk>\<And>x. \<lbrakk>x = polynate c; x = 0\<^sub>p\<rbrakk> \<Longrightarrow> P t; \<And>x. \<lbrakk>x = polynate c; x \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> P t\<rbrakk> \<Longrightarrow> P (CNP n c t)" shows "P a0"
is_regular_spair_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_regular_spair p q = (rep_list p \<noteq> 0 \<and> rep_list q \<noteq> 0 \<and> (let t1 = punit.lt (rep_list p); t2 = punit.lt (rep_list q); l = lcs t1 t2 in (l - t1) \<oplus> lt p \<noteq> (l - t2) \<oplus> lt q))"
simps_3: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and C1 :: "'a acom"   and C2 :: "'a acom" shows "Step f g S (C1;; C2) = Step f g S C1;; Step f g (post C1) C2"
sorted_insort_key: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
Fi: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> ZFC_in_HOL.set {j}" shows "F\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
less_eq_multiset_def: fixes M' :: "'b multiset"   and M :: "'b multiset" shows "(M' \<le> M) = (M' < M \<or> M' = M)"
prec_first_EV_WAIT_action: fixes aseq :: "int_point_t list"   and s :: "rstate_t"   and d :: "thread_id_t" assumes "is_SK_EV_WAIT aseq" shows "rprecondition s d (hd aseq)"
sheaf_spec_on_open_is_comm_ring: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and U :: "'d set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring (comm_ring.sheaf_spec R addition multiplication zero unit U) (comm_ring.add_sheaf_spec R addition multiplication zero U) (comm_ring.mult_sheaf_spec R addition multiplication zero U) (comm_ring.zero_sheaf_spec R addition multiplication zero unit U) (comm_ring.one_sheaf_spec R addition multiplication zero unit U)"
a_rcos_module_rev: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier G"   and "x' \<in> carrier G"   and "x' \<oplus> \<ominus> x \<in> H" shows "x' \<in> H +> x"
non_comm: shows "u\<^sub>0 \<cdot> u\<^sub>1 \<noteq> u\<^sub>1 \<cdot> u\<^sub>0"
Im_Ln_eq_pi_half_1: fixes z :: "complex" assumes "z \<noteq> 0" shows "(Im (Ln z) = pi / 2) = (0 < Im z \<and> Re z = 0)"
hyps_2: shows "c \<turnstile> n -CEdge (p', es, rets)\<rightarrow>\<^sub>p n'"
integrable_uniform_limit: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b" shows "f integrable_on cbox a b"
is_lasso_pre_empty: fixes G :: "('e, 'f) graph_rec_scheme"   and L :: "'e lasso" assumes "graph G"   and "g_V0 G = {}" shows "\<not> graph.is_lasso_pre G L"
interiorI: fixes T :: "'a set"   and x :: "'a"   and S :: "'a set" assumes "open T"   and "x \<in> T"   and "T \<subseteq> S" shows "x \<in> interior S"
assms_5: shows "\<forall>l op. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l \<and> l < length \<pi> \<longrightarrow> \<not> \<A> (Operator l (index (\<Pi>\<^sub>\<O>) op))"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> TargetTerm TQ"
simps_5: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
nn_integral_FTC_atLeast: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and F :: "real \<Rightarrow> real"   and T :: "real" assumes "f \<in> borel_measurable borel"   and "\<And>x. a \<le> x \<Longrightarrow> (F has_real_derivative f x) (at x)"   and "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"   and "(F \<longlongrightarrow> T) at_top" shows "\<integral>\<^sup>+x\<in>{a..}. ennreal (f x)\<partial>lborel = ennreal (T - F a)"
ma: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and nr :: "nat"   and n :: "nat"   and m1 :: "'b list list"   and nc :: "nat"   and m2 :: "'b list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI zer g f nr m1 m2 ! j ! i = scalar_prodI zer g f (row m1 i) (col m2 j)"
not_distinct_crossings_split_gallery: fixes Cs :: "'a set list" assumes "gallery Cs"   and "{} \<notin> set (wall_crossings Cs)"   and "\<not> distinct (wall_crossings Cs)" shows "\<exists>f g As A B Bs E F Fs. (f, g) \<in> foldpairs \<and> A \<in> f \<turnstile> \<C> \<and> B \<in> g \<turnstile> \<C> \<and> E \<in> g \<turnstile> \<C> \<and> F \<in> f \<turnstile> \<C> \<and> (Cs = As @ [A, B, F] @ Fs \<or> Cs = As @ [A, B] @ Bs @ [E, F] @ Fs)"
ct_suffix_cons_ct_list1: fixes xs :: "ct list"   and ys :: "t list"   and zs :: "t list" assumes "ct_suffix (xs @ to_ct_list ys) (to_ct_list zs)" shows "\<exists>as. zs = as @ ys \<and> ct_suffix xs (to_ct_list as)"
sum_length_filter_compl: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (filter P xs) + length (filter (\<lambda>x. \<not> P x) xs) = length xs"
simps_19: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'i \<Rightarrow> 'f"   and f3 :: "'j \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> 'f"   and f4 :: "('g, 'h, 'j, 'k) resumption \<Rightarrow> 'f"   and x31 :: "'j"   and x32 :: "'k \<Rightarrow> 'i" shows "(case Branch_corec x31 x32 of Terminal_corec x \<Rightarrow> f1 x | Linear_corec x xa \<Rightarrow> f2 x xa | Branch_corec x xa \<Rightarrow> f3 x xa | Resumption_corec x \<Rightarrow> f4 x) = f3 x31 x32"
CFG_intra_edge_no_Def_equal: fixes a :: "'edge"   and V :: "'var"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "intra_kind (kind a)"   and "pred (kind a) s" shows "state_val (transfer (kind a) s) V = state_val s V"
sp_no_jump: fixes F :: "'a \<Rightarrow> 'b fundef option"   and xs :: "('c, 'd, 'a, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr list"   and \<Sigma> :: "Unboxed.type option list"   and type :: "Unboxed.type option list" assumes "Subx.sp F xs \<Sigma> = Ok type" shows "\<forall>x\<in>set xs. \<not> is_jump x"
card_of_ordIso: fixes A :: "'d set"   and B :: "'e set" shows "(\<exists>f. bij_betw f A B) = ordIso2 (card_of A) (card_of B)"
eqvt: fixes p :: "perm"   and a :: "tm"   and aa :: "tm"   and ab :: "tm" shows "p \<bullet> PrfP a aa ab = PrfP (p \<bullet> a) (p \<bullet> aa) (p \<bullet> ab)"
real_sqrt_eq_1_iff: fixes x :: "real" shows "(sqrt x = 1) = (x = 1)"
degree_X: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>" shows "deg R (X_poly R) = 1"
real_transfer_operator_intg_2: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. f (T x) * g x)"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "LINT x|M. f x * real_transfer_operator g x = LINT x|M. f (T x) * g x"
transP_R2P: fixes r :: "('c \<times> 'c) set" shows "transP (R2P r) = trans r"
YM3_auth_B_to_A_lemma: fixes A :: "agent"   and nb :: "msg"   and B :: "agent"   and K :: "nat"   and NA :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>nb, Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA\<rbrace>, X\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>nb'. Says B Server \<lbrace>Agent B, nb', Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs"
fact_in_Reals: fixes n :: "nat" shows "fact n \<in> \<real>"
invariant_initE: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and s :: "'e" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "s \<in> init A" shows "P s"
simps_30: fixes x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp"   and x31 :: "nat"   and x32 :: "exp" shows "EIf x61 x62 x63 \<noteq> ELam x31 x32"
dij_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" shows "dij a b i j = lcm (a ! i) (b ! j) div a ! i"
integrable_I0i_exp_mscale: fixes u :: "real" assumes "0 < u" shows "set_integrable lborel {0<..} (\<lambda>x. exp (- (x * u)))"
lm033: fixes a :: "('b \<times> 'a set) set" assumes "runiq (a\<inverse>)"   and "runiq a"   and "finite a"   and "{} \<notin> Range a"   and "\<forall>pair\<in>a. finite (snd pair)" shows "card (pseudoAllocation a) = sum card (Range a)"
Ints_nonzero_abs_less1: fixes x :: "'a" assumes "x \<in> \<int>"   and "\<bar>x\<bar> < (1::'a)" shows "x = (0::'a)"
le_utrans_trans: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and z :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_utrans x y"   and "le_utrans y z" shows "le_utrans x z"
extensional_funcset_def: fixes S :: "'d set"   and T :: "'e set" shows "S \<rightarrow>\<^sub>E T = (S \<rightarrow> T) \<inter> extensional S"
simps_3: fixes x21 :: "'a"   and x22 :: "'a list" shows "Cons x21 x22 \<noteq> []"
b1: shows "Im (infsetsum g A) = (\<Sum>\<^sub>ax\<in>A. Im (g x))"
bullet_app_redL: fixes t :: "CL"   and t' :: "CL"   and u :: "CL" assumes "(t, t') \<in> red" shows "(bullet_app t u, bullet_app t' u) \<in> red\<^sup>*"
subsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "order.greater_eq B A"   and "c \<in> A" shows "c \<in> B"
learn_cons_closed_subseteq: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and V :: "(nat \<Rightarrow> nat option) set" assumes "learn_cons \<psi> U s"   and "V \<subseteq> U" shows "learn_cons \<psi> V s"
i_le_j_events_neq1: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "i < j"   and "j < card X"   and "f j \<noteq> b" shows "f i \<noteq> f j"
feasible_subpaths_preserved: fixes rv :: "'a \<times> nat" assumes "RedBlack prb"   and "rv \<in> red_vertices prb" shows "feasible_subpaths_from (black prb) (confs prb rv) (fst rv) \<subseteq> RedBlack_subpaths_from prb rv"
dist_fmap_def: fixes P :: "'d \<Rightarrow>\<^sub>F 'e"   and Q :: "'d \<Rightarrow>\<^sub>F 'e" shows "dist P Q = (MAX i. dist ((P)\<^sub>F i) ((Q)\<^sub>F i)) + (if domain P = domain Q then 0 else 1)"
REMOVE_INTERNAL_def: fixes x :: "'e" shows "REMOVE_INTERNAL x \<equiv> x"
no_self_edges: fixes g :: "graph"   and a :: "nat" assumes "minGraphProps' g" shows "(a, a) \<notin> \<E> g"
standard_simplex_01: fixes p :: "nat" shows "standard_simplex p \<subseteq> UNIV \<rightarrow>\<^sub>E {0..1}"
intros_1: fixes xs :: "nat list"   and ys :: "nat list" assumes "length xs = length ys"   and "xs \<noteq> ys" shows "Form 0 {xs, ys}"
body_rewrite_fundef_body: fixes fd :: "'a fundef"   and n :: "nat"   and x :: "'a" shows "body (rewrite_fundef_body fd n x) = rewrite (body fd) n x"
finite_imageI: fixes F :: "'f set"   and h :: "'f \<Rightarrow> 'g" assumes "finite F" shows "finite (h ` F)"
simps: fixes a1 :: "'vtx"   and a2 :: "'vtx"   and a3 :: "('vtx \<times> 'lbl \<times> 'vtx) list" shows "path a1 a2 a3 = ((\<exists>l1 l2. a1 = l1 \<and> a2 = l2 \<and> a3 = [] \<and> l1 = l2) \<or> (\<exists>l1 l2 xs lbl l3. a1 = l1 \<and> a2 = l3 \<and> a3 = xs @ [(l2, lbl, l3)] \<and> path l1 l2 xs \<and> lbl \<in>\<^sub>A weights l2 l3))"
assms_2: shows "2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2)"
digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "digraph G \<equiv> fin_digraph G \<and> loopfree_digraph G \<and> nomulti_digraph G"
lm012: fixes G :: "'c set"   and N :: "'d set" shows "possible_allocations_rel G N \<subseteq> {a. Range a \<subseteq> N \<and> Domain a \<in> all_partitions G}"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Cod (t \<^bold>\<cdot> u) = Cod t"
joinable_Nil_r_iff: fixes q :: "'a list" shows "joinable q [] = False"
rt_graph_def: fixes \<sigma> :: "nat \<Rightarrow> state" shows "rt_graph \<sigma> = (\<lambda>dip. {uu_. \<exists>ip ip' dsn dsk hops pre. uu_ = (ip, ip') \<and> ip \<noteq> dip \<and> rt (\<sigma> ip) dip = Some (dsn, dsk, val, hops, ip', pre)})"
lookup_pair_eq_value: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and v :: "'b"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "v \<noteq> (0::'b)" shows "(comparator.lookup_pair comp xs k = v) = ((k, v) \<in> set xs)"
pr_strict_asymp: fixes weights :: "'a weights" assumes "class.weighted weights" shows "asymp (pr_strict weights)"
Sura_Bura_clopen_subset_alt: fixes S :: "'a set"   and C :: "'a set"   and U :: "'a set" assumes "locally compact S"   and "C \<in> components S"   and "compact C"   and "openin (top_of_set S) U"   and "C \<subseteq> U" obtains K :: "'a set" where "openin (top_of_set S) K" and "compact K" and "C \<subseteq> K" and "K \<subseteq> U"
ball_single: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>{x}. P x) = P x"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "src \<mu> = src f"
IH_1: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "lower_on \<sigma> \<eta> (vars_of t1)" shows "lower_or_eq (t1 \<lhd> \<sigma>) (t1 \<lhd> \<eta>)"
zip_map_option1: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e option"   and y :: "'d option" shows "zip_option (map_option f x) y = map_option (apfst f) (zip_option x y)"
Nmlize_Trg_1: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Trg t\<^bold>\<rfloor> = Trg \<^bold>\<lfloor>t\<^bold>\<rfloor>"
c: shows "(\<lambda>_. 0\<^sub>v n) \<in> {0..<n} \<rightarrow> carrier_vec n"
hyps: fixes n :: "Integer" assumes "le\<cdot>1\<cdot>n = TT"   and "le\<cdot>(length\<cdot>xs + n)\<cdot>0 = TT" shows "False"
conc_UNION_distrib_2: fixes M :: "'c \<Rightarrow> 'b list set"   and I :: "'c set"   and A :: "'b list set" shows "\<Union> (M ` I) @@ A = (\<Union>i\<in>I. M i @@ A)"
G2: shows "geodesic_segment_between G2 (g a) (g (dist x y))"
map_id: fixes t :: "'b \<times> 'c" shows "map_prod id id t = t"
assms: shows "x \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b)"
sig_red_zero_subset: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red_zero sing_reg F p"   and "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)" shows "sig_red_zero sing_reg {f \<in> F. sing_reg (lt f) (lt p)} p"
in_arcs_G: shows "{uv, rev_G uv} \<subseteq> arcs G"
add_eq_0_iff2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (a = uminus b)"
finite_abs_reach: fixes f :: "'b ltl \<Rightarrow> 'c \<Rightarrow> 'b ltl"   and \<phi> :: "'b ltl" assumes "lift_ltl_transformer f" shows "finite (lift_ltl_transformer.abs_reach f (Abs \<phi>))"
set_of_mul_inc_right: fixes B :: "'b interval"   and B' :: "'b interval"   and A :: "'b interval" assumes "set_of B \<subseteq> set_of B'" shows "set_of (A * B) \<subseteq> set_of (A * B')"
a: shows "f \<otimes> foldr (\<otimes>) fs' \<one> \<sim> a"
not_le: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
intros_7: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list"   and V :: "'c"   and x :: "nat"   and n :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs"   and "V = ins ! x"   and "x < length ins"   and "n = Actual_in (sourcenode a, x)"   and "n' = Formal_in (targetnode a, x)" shows "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (Some V) (Some (p, True)) n'"
fair_priceI: fixes Mkt :: "('a, 'c) discrete_market"   and \<pi> :: "real"   and pyf :: "'a \<Rightarrow> real"   and matur :: "nat" assumes "fair_price Mkt \<pi> pyf matur" shows "\<exists>pr. price_structure pyf matur \<pi> pr \<and> (\<forall>x. x \<notin> stocks Mkt \<longrightarrow> (\<forall>Mkt2 p. coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr \<and> portfolio p \<and> support_set p \<subseteq> stocks Mkt \<union> {x} \<longrightarrow> \<not> arbitrage_process Mkt2 p))"
trans_converse: fixes r :: "('b \<times> 'b) set" shows "trans (r\<inverse>) = trans r"
IH_2: shows "State_Monad.ap (State_Monad.return dlabels) (label_tree r) = symbols (leaves r)"
cut_less_cut_ge_ident: fixes t2 :: "'a"   and t1 :: "'a"   and I :: "'a set" assumes "t2 \<le> t1" shows "I \<down>< t1 \<union> I \<down>\<ge> t2 = I"
bv_nat_bv: fixes n :: "nat" shows "bv_to_nat (nat_to_bv n) = n"
extendEmpty: fixes C :: "'a sequent" shows "extend ( \<Empt> \<Rightarrow>* \<Empt>) C = C"
assms_2: shows "Fun f (map Var Z) \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t subst_range \<delta>"
Field_set_code: fixes rel :: "('a \<times> 'a) list" shows "Field (set rel) = set (exefield rel)"
unit_disc_fix_preserve_poincare_collinear_iff: fixes M :: "moebius"   and A :: "complex_homo set" assumes "unit_circle_fix M" shows "poincare_collinear (moebius_pt M ` A) = poincare_collinear A"
simps_1: fixes z :: "complex" shows "ReZ (round_complex z) = round (Re z)"
ncc_def: fixes X :: "'a set" shows "ncc X = (X \<noteq> {} \<and> compact X \<and> convex X)"