hcju/isabelleps huggingface dataset:
is_unifiers_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma> :: "'c"   and AAA :: "'b set set" shows "substitution_ops.is_unifiers subst_atm \<sigma> AAA = (\<forall>AA\<in>AAA. substitution_ops.is_unifier subst_atm \<sigma> AA)"
poly_div_gcd_squarefree_aux_2: fixes p :: "'a poly"   and x :: "'a" assumes "pderiv p \<noteq> 0" shows "(poly (p div gcd p (pderiv p)) x = (0::'a)) = (poly p x = (0::'a))"
lenlex_length: fixes ms :: "'g list"   and ns :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(ms, ns) \<in> lenlex r" shows "length ms \<le> length ns"
assoc: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
arbitrage_processE: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "disc_filtr_prob_space M F"   and "disc_filtr_prob_space.arbitrage_process M F Mkt p" shows "\<exists>m. self_financing Mkt p \<and> disc_filtr_prob_space.trading_strategy F p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < \<P>(w in M. 0 < cls_val_process Mkt p m w)"
is_dcdi_via_def: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and v :: "'var"   and m :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n"   and m' :: "nat" shows "(n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m') = (is_path \<pi> \<and> m < n \<and> (\<exists>l' n'. cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m' \<and> cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n' \<and> n' dd\<^bsup>\<pi>',v\<^esup>\<rightarrow> l' \<and> l' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> m') \<and> (\<forall>l\<in>{m..<n}. v \<notin> local.writes (\<pi> l)))"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = max a b"
bij_betw_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and B :: "'a set"   and B' :: "'b set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw f B B'"
mono: fixes c :: "'d"   and a :: "'d"   and d :: "'d"   and b :: "'d" assumes "c \<le> a"   and "d \<le> b" shows "sup c d \<le> sup a b"
Spec: 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 r :: "('e \<times> 'e) set"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)} Spec r Q,A"
simps_2: fixes I :: "('e, 'f, 'd) interp"   and f :: "'e"   and args :: "'d \<Rightarrow> ('e, 'd) trm"   and v :: "(real, 'd) vec" shows "frechet I ($f f args) v = (\<lambda>v'. FunctionFrechet I f (\<chi>i. sterm_sem I (args i) v) (\<chi>i. frechet I (args i) v v'))"
prems: shows "\<forall>fm'. ListMem fm' (a # fmList) \<longrightarrow> agree fm fm'"
GuaranteeAsGuard: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> g \<inter> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P guaranteeStrip f g c Q,A"
simps: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap" shows "ArityAnalysisLetSafe.a_consistent Aexp a1 a2 = (\<exists>ae \<Gamma> S a e as. a1 = (ae, a, as) \<and> a2 = (\<Gamma>, e, S) \<and> edom ae \<subseteq> domA \<Gamma> \<union> upds S \<and> Astack S \<sqsubseteq> a \<and> (ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>ae \<squnion> Aexp e\<cdot>a \<squnion> ArityAnalysis.AEstack Aexp as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae \<and> stack_consistent as S)"
Y: shows "Y = Im (2 * z / (1 + z * cnj z))"
M_PVar: fixes T :: "POPLmarkRecord.type"   and t :: "trm" shows "\<turnstile> PVar T \<rhd> t \<Rightarrow> [t]"
comp_arr_ide: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VV.ide a"   and "H.VV.seq f a" shows "H.VV.comp f a = f"
E_b1_c: shows "LeftDerivation b1 E (take j c)"
phi_5: shows "bij_betw \<phi> {0..1} {0..1}"
finfun_Diag_const1: fixes b :: "'b"   and g :: "('a, 'c) finfun" shows "finfun_Diag (finfun_const b) g = finfun_comp (Pair b) g"
call_borel: fixes T :: "nat"   and K :: "real" shows "call_option T K \<in> borel_measurable (G T)"
Deduct_axioms: shows "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi bprv"
map_int_tNN: fixes Tl :: "'fsym trm list"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "list_all Ik.wt Tl"   and "Ik.wtE \<xi>" shows "map2 ntsem (map Ik.tpOf Tl) (map (Ik.int \<xi>) Tl) = map (TE.int (eenv \<xi>) \<circ> tNN) Tl"
intro: fixes P :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>a b. \<lbrakk>a \<in> P; b \<in> P; less_eq a b\<rbrakk> \<Longrightarrow> less_eq' (f a) (f b)" shows "OrderingSetMap_axioms less_eq less_eq' P f"
same_prefix_nil: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<cdot> ys \<le>p xs) = (ys = \<epsilon>)"
Im_Ln_le_pi: fixes z :: "complex" assumes "z \<noteq> 0" shows "Im (Ln z) \<le> pi"
eq1: shows "lookup (p * (monomial c (pp_of_term v) when component_of_term v = component_of_term u)) (pp_of_term u) = (lookup (p * monomial c (pp_of_term v)) (pp_of_term u) when component_of_term v = component_of_term u)"
curry_uncurry_id: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "curry (uncurry f) = f"
of_nat_nat: fixes z :: "int" assumes "0 \<le> z" shows "of_nat (nat z) = of_int z"
\<Omega>_cover: fixes i :: "'a" assumes "i \<in> I" shows "\<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union> S"
in_doubt_allow_allows_Accept: fixes a :: "action"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and m :: "'c match_expr"   and p :: "'d" assumes "a = action.Accept"   and "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "Semantics.matches \<gamma> m p" shows "Matching_Ternary.matches (\<beta>, in_doubt_allow) m a p"
open_kriterion: fixes t :: "'a set" assumes "\<And>x. x \<in> t \<Longrightarrow> \<exists>t'. t' open \<and> x \<in> t' \<and> t' \<subseteq> t" shows "t open"
Beta: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list"   and t :: "dB" assumes "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>l t" shows "Abs r \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
one_dim_clinear_eqI: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x \<noteq> (0::'a)"   and "clinear f"   and "clinear g"   and "f x = g x" shows "f = g"
False: shows "\<not> (\<forall>k<e_length e. \<psi> i k \<down>)"
subdivision_pair_base: fixes G :: "'b pair_pre_digraph" assumes "pair_bidirected_digraph G" shows "subdivision_pair G G"
simps_1: fixes x :: "brick list" shows "codomain_wall (basic x) = codomain_block x"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "boa x y"   and "boa y z"   and "x \<in> A" shows "boa x z"
fundantivertex: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "fundantivertex s \<in> C0 - s `\<rightarrow> C0"
state_fun_def: fixes f :: "'sf" shows "state_fun f = $f f Var"
dom_char: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "local.dom f = (if arr f then MkIde (Dom f) else null)"
cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'a" and xs :: "'a list" where "x = xa # xs"
times_eint_def: fixes m :: "eint"   and n :: "eint" shows "m * n = (case m of eint m \<Rightarrow> case n of eint n \<Rightarrow> eint (m * n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
Collect_mem_eq: fixes A :: "'b::type set" shows "{x::'b::type. x \<in> (A::'b::type set)} = A"
eq_fp64_def: shows "eq_fp64 \<equiv> (=)"
distinct_no_loop2: fixes f :: "face"   and v :: "nat"   and u :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "u \<in> \<V> f"   and "u \<noteq> v" shows "f \<bullet> v \<noteq> v"
pred_cong: fixes x :: "'d list"   and ya :: "'d list"   and P :: "'d \<Rightarrow> bool"   and Pa :: "'d \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> P z = Pa z" shows "list_all P x = list_all Pa ya"
ivp2: shows "Y\<^sub>2 \<in> Sols f U S t\<^sub>0 s"
measure_spmf_eq_density: fixes p :: "'c spmf" shows "measure_spmf p = density (count_space UNIV) (\<lambda>x. ennreal (spmf p x))"
Quotient_ltln\<^sub>L: shows "Quotient (\<sim>\<^sub>L) abs_ltln\<^sub>L rep_ltln\<^sub>L cr_ltln\<^sub>L"
substPB_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'a trm"   and \<Phi> :: "('a, 'b) lit list set" shows "substPB \<pi> \<Phi> = {substC \<pi> c |c. c \<in> \<Phi>}"
f'_def: shows "f' \<equiv> \<lambda>\<rho> x. f \<rho> (op_sem Add <|x, expr_sem_rf \<rho> (Minus $$ e')|>)"
trace_preserves_unowned_shared: fixes c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "simple_ownership_distinct (fst (c n))"   and "unowned_shared (snd (snd (c n))) (fst (c n))"   and "trace c n k"   and "\<forall>x<k. safe_delayed (c (n + x))"   and "l \<le> k" shows "unowned_shared (snd (snd (c (n + l)))) (fst (c (n + l)))"
essential_classI: fixes C :: "'s set" assumes "C \<in> UNIV // communicating"   and "\<And>x y. \<lbrakk>x \<in> C; (x, y) \<in> acc\<rbrakk> \<Longrightarrow> y \<in> C" shows "essential_class C"
solvable_def: fixes G :: "('a, 'b) monoid_scheme" shows "solvable G = solvable_seq G (carrier G)"
simps_17: shows "cospan_shape.comp cospan_shape.AT cospan_shape.Null = cospan_shape.Null"
induct: fixes P1 :: "POPLmarkRecord.type \<Rightarrow> bool"   and P3 :: "(char list \<times> POPLmarkRecord.type) list \<Rightarrow> bool"   and P2 :: "char list \<times> POPLmarkRecord.type \<Rightarrow> bool"   and char_list_type_prod_list :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<And>x. P1 (TVar x)"   and "P1 Top"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<rightarrow> x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (\<forall><:x1. x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (RcdT x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P3 char_list_type_prod_list"
\<alpha>g_def: fixes g :: "'m" shows "\<alpha>g g \<equiv> graph (\<alpha>nodes_aux g) (\<alpha>edges_aux g)"
hyps: shows "ys \<noteq> [] \<and> (\<forall>x\<in>set ys. x < m) \<longrightarrow> sum_list ys < m * real (length ys)"
intros: fixes q :: "'d"   and W :: "'d set"   and dsqr :: "('d, 'e) ta_rule set"   and \<delta> :: "('d, 'e) ta_rule set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and Q' :: "'d \<Rightarrow> 'e tree option"   and Q :: "'d \<Rightarrow> 'e tree option"   and W' :: "'d set"   and rcm' :: "('d, 'e) ta_rule \<Rightarrow> nat option" assumes "q \<in> W"   and "dsqr = {r \<in> \<delta>. q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1}"   and "dom Q' = dom Q \<union> lhs ` dsqr"   and "\<And>q t. Q' q = Some t \<Longrightarrow> Q q = Some t \<or> (\<exists>r\<in>dsqr. q = lhs r \<and> t = construct_witness Q r)"   and "W' = W - {q} \<union> (lhs ` dsqr - dom Q)"   and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm' r = (if q \<in> set (rhsq r) then Some (the (rcm r) - 1) else rcm r)" shows "((Q, W, rcm), Q', W', rcm') \<in> brw_step \<delta>"
th0: shows "(fps_deriv a oo fps_inv a) $ 0 \<noteq> (0::'a)"
\<alpha>\<beta>_refl: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" shows "s \<approx> s"
i_set_mult_div_self_i_set: fixes I :: "nat set"   and k :: "nat" assumes "I \<in> i_set_mult k" shows "I \<oslash> k \<in> i_set"
Ints_nonzero_abs_ge1: fixes x :: "'a" assumes "x \<in> \<int>"   and "x \<noteq> (0::'a)" shows "(1::'a) \<le> \<bar>x\<bar>"
int_nat_p_pow_minus: fixes k :: "nat" shows "int (nat (p ^ k - 1)) = p ^ k - 1"
semIntAbs_prSubstAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSubstAbs (semInt SEM) (semIntAbs SEM) SEM"
wrap_card: fixes B :: "'b set" shows "card (wrap B) \<le> 1"
f_Exec_Stream_Init_nth: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n \<le> length xs" shows "f_Exec_Comp_Stream_Init trans_fun xs c ! n = f_Exec_Comp trans_fun (xs \<down> n) c"
assms_1: shows "c \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. ASSERT \<Psi> \<bind> (\<lambda>_. m))"
find_base_vectors_def: fixes uminus :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and A :: "'b mat" shows "find_base_vectors_gen uminus zero one A \<equiv> let pp = pivot_positions_gen zero A in Let (filter (\<lambda>j. j \<notin> set (map snd pp)) [0..<dim_col A]) (map (non_pivot_base_gen uminus zero one A pp))"
dom_rhs: shows "\<D>\<^sub>\<circ> [\<CC>\<lparr>Obj\<rparr>, \<CC>\<lparr>Arr\<rparr>, \<CC>\<lparr>Dom\<rparr>, \<CC>\<lparr>Cod\<rparr>]\<^sub>\<circ> = 4\<^sub>\<nat>"
InitNoneRed: fixes sh :: "sheap"   and C :: "char list"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = None" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh(C \<mapsto> (sblank P C, Prepared))),b\<rangle>"
PermI: fixes Gamma' :: "formula list"   and R :: "(formula list \<times> formula list set) set"   and Gamma :: "formula list" assumes "Gamma' \<in> deductions R"   and "Gamma <~~> Gamma'"   and "Perms \<subseteq> R" shows "Gamma \<in> deductions R"
scaleR: fixes r :: "real"   and b :: "(real, 'n) vec" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
coeff_inject: fixes x :: "'b poly"   and y :: "'b poly" shows "(coeff x = coeff y) = (x = y)"
extensibleD: fixes R :: "('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" assumes "extensible R G f" obtains g :: "('c \<times> 'd) set" where "graph_homomorphism (snd R) G g" and "agree_on (fst R) f g"
card_Ex_subset: fixes k :: "nat"   and M :: "'a set" assumes "k \<le> card M" shows "\<exists>N\<subseteq>M. card N = k"
constant_functor_axioms: shows "constant_functor J.comp (\<cdot>) a"
adjoint_safe: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "is_interp (local.adjoint I \<sigma> \<nu>)"
eSuc_mono: fixes n :: "eint"   and m :: "eint" shows "(n + 1 < m + 1) = (n < m)"
prems_1: shows "vs @ xs' @ [x] \<in> C M2 M1 \<Omega> V m i"
H1: shows "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall>X<:T11. T12)"
ddh_0_def: fixes \<G> :: "'d cyclic_group"   and \<A> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool spmf" shows "Diffie_Hellman.ddh.ddh_0 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> (x * y))))"
step: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and n :: "nat" assumes "\<not> P \<omega>"   and "ev_at P n (stl \<omega>)" shows "ev_at P (Suc n) \<omega>"
select_convs_2: fixes es :: "'d"   and ps :: "'a \<Rightarrow> 'b"   and pubActs :: "'c \<times> ('a \<Rightarrow> 'e)"   and more :: "'f" shows "ps \<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr> = ps"
ins_in_Def: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs"   and "V \<in> set ins" shows "V \<in> Def (targetnode a)"
dghm_singleton_ArrMap_vsv: fixes j :: "V"   and \<CC> :: "V" shows "vsv (dghm_singleton j \<CC>\<lparr>ArrMap\<rparr>)"
zero_less_iff_neq_zero: fixes n :: "'a multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
dg_Set_is_arrD_2: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B" shows "S\<lparr>ArrDom\<rparr> = A"
psi_inner_psi'_l: fixes l :: "nat" shows "inner_prod \<psi> (psi'_l l) = alpha_l l * complex_of_real (cos (\<theta> / 2)) - beta_l l * complex_of_real (sin (\<theta> / 2))"
distributed_borel_measurable: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> ennreal" assumes "distributed M N X f" shows "f \<in> borel_measurable N"
k0: fixes x :: "'c" shows "k (0, x) = g x"
time_find_closest_pair_tm: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" shows "time (find_closest_pair_tm (c\<^sub>0, c\<^sub>1) ps) \<le> 17 * length ps + 1"
lprefix_cons1_def: fixes x :: "'a"   and sg' :: "'sg"   and sh :: "'sh" shows "local.lprefix_cons1 x sg' sh = lprefix (LCons x (lunstream g sg')) (lunstream h sh)"
mk_poly2_row: fixes n :: "nat"   and pv :: "'b poly vec"   and A :: "'b mat"   and d :: "nat"   and j :: "nat"   and i :: "nat" assumes "n \<le> dim_vec pv" shows "mk_poly2_row A d j pv n $ i = (if i < n then pv $ i + monom (A $$ (i, j)) d else pv $ i)"
pivot_fun_eliminate_entries: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and jj :: "nat"   and nr :: "nat"   and nc :: "nat"   and l :: "nat"   and vs :: "nat \<Rightarrow> 'c"   and j :: "nat" assumes "pivot_fun A f jj"   and "dim_row A = nr"   and "dim_col A = nc"   and "f l = jj"   and "l < nr"   and "jj \<le> nc" shows "pivot_fun (eliminate_entries vs A l j) f jj"
path_join_imp: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" assumes "path g1"   and "path g2"   and "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2)"
polyfun_extremal: fixes c :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat"   and B :: "real" assumes "c k \<noteq> (0::'a)"   and "1 \<le> k"   and "k \<le> n" shows "\<forall>\<^sub>F z in at_infinity. B \<le> norm (\<Sum>i\<le>n. c i * z ^ i)"
last_message_exists_nth: fixes xs :: "'b message_af list" assumes "last_message xs \<noteq> NoMsg" shows "\<exists>i<length xs. last_message xs = xs ! i \<and> (\<forall>j<length xs. i < j \<longrightarrow> xs ! j = NoMsg)"
assms_2: shows "(g \<longlongrightarrow> b) F"
generating_ring: shows "ring_of_sets \<Omega> S.generated_ring"
smallest_accepting_rank\<^sub>\<R>_properties: fixes i :: "nat" assumes "\<MM>.smallest_accepting_rank\<^sub>\<R> = Some i" shows "accepting_pair\<^sub>R \<MM>.\<delta>\<^sub>\<R> \<MM>.q\<^sub>\<R> (\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R i, \<MM>.succeed\<^sub>R i) w"
prems_3: shows "mod_type_class.to_nat a \<le> Suc n"
lookup_update: fixes k :: "'b"   and v :: "'a"   and m :: "('b, 'a) mapping" shows "Mapping.lookup (Mapping.update k v m) k = Some v"
atoms_formula_simplify: fixes S :: "'at Literal set set" shows "atoms_formula (simplify S) \<subseteq> atoms_formula S"
bij_image_Compl_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "bij f" shows "f ` (- A) = - f ` A"
fls_deriv_residue: fixes f :: "'a fls" shows "fls_nth (fls_deriv f) (- 1) = (0::'a)"
CTXT_def: fixes inp :: "nat"   and ct :: "(char list \<times> nat \<times> unit list) list"   and outp :: "nat"   and P :: "'a" shows "Case_Labeling.CTXT inp ct outp P \<equiv> P"
groupF_cong: fixes xs :: "'a list"   and f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'c" assumes "\<forall>x\<in>set xs. \<forall>y\<in>set xs. (f1 x = f1 y) = (f2 x = f2 y)" shows "groupF f1 xs = groupF f2 xs"
not_arr_null: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "partial_magma C" shows "\<not> partial_magma.arr C (partial_magma.null C)"
eval_elim_cases_12: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and T :: "ref_ty"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (e InstOf T)\<succ>\<rightarrow> (v, s')" obtains va :: "val" where "v = In1 (Bool (va \<noteq> Null \<and> G,snd s'\<turnstile>va fits RefT T))" and "G\<turnstile>Norm s \<midarrow>e-\<succ>va\<rightarrow> s'"
agsum_add2m: fixes a :: "'a"   and n :: "int" assumes "a \<in> carrier A"   and "n < 0" shows "\<zero> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> (- n)\<triangleright>a\<^bsub>A\<^esub>"
fd_def: fixes x :: "'a"   and y :: "'a" shows "\<langle>x| y = ar ar (y \<cdot> x)"
assms: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "1 < degree p" shows "\<exists>x\<in>carrier R. local.eval p x = \<zero>"
smult_0_right: fixes a :: "'b" shows "smult a 0 = 0"
c_def: shows "c \<equiv> complex_of_real (2 * pi) * \<i>"
complete_linorder_inf_min: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "inf = ord.min less_eq"
odd_real_root_unique: fixes n :: "nat"   and y :: "real"   and x :: "real" assumes "odd n"   and "y ^ n = x" shows "root n x = y"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> left.transformation.orbit.Partition" shows "left.transformation.orbit.Class a = A"
\<phi>_in_hom_1: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>" shows "\<guillemotleft>\<phi> \<mu> : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
anonymous': fixes R1 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and R2 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "anonymous_profile R1 = anonymous_profile R2"   and "is_pref_profile R1"   and "is_pref_profile R2" shows "scf R1 = scf R2"
simps_13: fixes b :: "nat"   and I :: "\<I>"   and \<phi> :: "Formula.formula" shows "Formula.fvi b (formula.Next I \<phi>) = Formula.fvi b \<phi>"
ennreal_suminf_minus: fixes g :: "nat \<Rightarrow> ennreal"   and f :: "nat \<Rightarrow> ennreal" assumes "\<And>i. g i \<le> f i"   and "suminf f \<noteq> top"   and "suminf g \<noteq> top" shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
strategyproofness: shows "pmf (sds R30) b + pmf (sds R30) a \<le> pmf (sds R47) b + pmf (sds R47) a"