hcju/isabelleps huggingface dataset:
ldropn_Suc_LCons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b llist" shows "ldropn (Suc n) (LCons x xs) = ldropn n xs"
dvd_minus_mod: fixes b :: "'a"   and a :: "'a" shows "b dvd a - a mod b"
reachable_nodes_refl: fixes r :: "'b"   and g :: "'b ugraph" shows "r \<in> reachable_nodes g r"
P1: shows "A \<noteq> X \<and> A \<noteq> Z \<and> \<not> Col Y A X \<and> \<not> Col Z A X \<and> \<not> Col Y A Z"
prems_3: shows "Im (trace (complex_of_real c \<cdot>\<^sub>m \<rho>)) = 0"
1: fixes ars :: "String.literal \<Rightarrow> String.literal set list option"   and a :: "String.literal"   and b :: "String.literal"   and y :: "String.literal set list" assumes "ars \<in> ran (map_of (map (apsnd map_of) arss))"   and "(a, b) \<in> set cs"   and "ars a = Some y" shows "\<exists>y. ars b = Some y"
map: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'c"   and x :: "'e" shows "map_phantom f1 f2 (Phantom('d) x) = Phantom('b) (f2 x)"
all_roots_real_degree: fixes p :: "real poly" assumes "all_roots_real p" shows "proots_count p UNIV = degree p"
configuration_step: fixes x :: "nat"   and q' :: "'b"   and n :: "nat"   and q :: "'b" assumes "x \<in> configuration q' n"   and "q = \<delta> q' (w n)" shows "x \<in> configuration q (Suc n)"
arr_char: fixes C1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<times> 'c" assumes "product_category C1 C2" shows "partial_magma.arr (product_category.comp C1 C2) f = (partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f))"
affine_independent_2: fixes a :: "'b"   and b :: "'b" shows "\<not> affine_dependent {a, b}"
assms_1: shows "Deliver m2 \<in> set (history j)"
less_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "(i1 < i2) = (i1 \<le> i2 \<and> \<not> i2 \<le> i1)"
3: shows "Well_order r''' \<and> Field r''' = B - f ` A"
add_inf_1: fixes a :: "'a DBMEntry" shows "a + \<infinity> = \<infinity>"
u_nonneg: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "0 \<le> u x y n"
toInteger_Integer_def: shows "toInteger = (\<Lambda> x. x)"
as_2: shows "\<forall>y. y \<in> s \<longrightarrow> y = x"
simps_1: fixes p :: "char list"   and ts :: "hterm list" shows "flit_of_hlit (Pos p ts) = Pos p (fterms_of_hterms ts)"
satisfies_T0: shows "is_left_adjoint r\<^sub>0"
IH2: assumes "OUadmitFO \<sigma> ODE2 U" shows "osafe (OsubstFO ODE2 \<sigma>)"
g: shows "Sorted_Less.sorted (rev \<tau>) \<and> Sorted_Less.sorted (rev \<tau>') \<and> set \<tau>' \<subseteq> set \<tau> \<or> \<tau> = \<tau>'"
hyps_2: shows "a \<inter> {m. domain m \<in> J} \<in> sets (Pi\<^sub>F J M)"
af_letter_abs_opt_def: shows "\<up>af\<^sub>\<UU> \<equiv> af_abs_opt.f_abs"
long_ch_by_ord_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.long_ch_by_ord betw f X \<equiv> \<exists>x\<in>X. \<exists>y\<in>X. \<exists>z\<in>X. x \<noteq> y \<and> y \<noteq> z \<and> x \<noteq> z \<and> TernaryOrdering.ordering f betw X"
rk_singleton: shows "\<forall>P. rk {P} = 1"
power_0: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" shows "power.power one times a 0 = one"
bot_fset_def: shows "{||} \<equiv> Abs_fset {}"
R12: fixes x :: "'M" shows "0 < R x \<and> R x \<le> 1 / 2"
oreceived_msg_inv: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> msg \<Rightarrow> bool"   and Q :: "state \<Rightarrow> state \<Rightarrow> bool"   and i :: "nat" assumes "\<And>\<sigma> \<sigma>' m. \<lbrakk>P \<sigma> m; other Q {i} \<sigma> \<sigma>'\<rbrakk> \<Longrightarrow> P \<sigma>' m"   and "\<And>\<sigma> m. P \<sigma> m \<Longrightarrow> P (\<sigma>(i := \<sigma> i\<lparr>msg := m\<rparr>)) m" shows "optoy i \<Turnstile> (otherwith Q {i} (orecvmsg P), other Q {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). l \<in> {PToy-:1} \<longrightarrow> P \<sigma> (msg (\<sigma> i)))"
ordIso_ordLess_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r =o r'"   and "r' <o r''" shows "r <o r''"
arith_geo_mean: fixes u :: "'a"   and x :: "'a"   and y :: "'a" assumes "u\<^sup>2 = x * y"   and "(0::'a) \<le> x"   and "(0::'a) \<le> y" shows "u \<le> (x + y) / (2::'a)"
intro: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat" assumes "abstract_LP A b c m n" shows "rat_LP A b c m n"
round_down_def: fixes prec :: "int"   and x :: "real" shows "round_down prec x = real_of_int \<lfloor>x * 2 powr real_of_int prec\<rfloor> * 2 powr - real_of_int prec"
weakenS: fixes \<sigma> :: "'e"   and \<sigma>' :: "'e"   and a :: "'f" assumes "S \<sigma> \<sigma>' a" shows "S' \<sigma> \<sigma>' a"
weq: shows "ennreal (weight_spmf p) = ennreal (weight_spmf q)"
rep_eq: fixes x :: "('a, 'b) fmap"   and xa :: "('a, 'b) fmap" shows "fmlookup (x ++\<^sub>f xa) = Map.map_add (fmlookup x) (fmlookup xa)"
unique_factorization_m_alt_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "unique_factorization_m f cfs = (factorization_m f cfs \<and> (\<forall>dgs. factorization_m f dgs \<longrightarrow> Mf dgs = Mf cfs))"
refine: shows "(impl_red_dfs_ahs, red_dfs) \<in> \<langle>Id\<rangle>slg_rel \<rightarrow> \<langle>Id\<rangle>ahs.rel \<rightarrow> \<langle>Id\<rangle>ahs.rel \<rightarrow> Id \<rightarrow> \<langle>\<langle>Id\<rangle>ahs.rel \<times>\<^sub>r red_wit_rel Id\<rangle>nres_rel"
A: shows "f \<in> {..n} \<rightarrow> carrier G"
is_backward_sim_def: fixes f :: "'b \<Rightarrow> 'c set"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa" shows "is_backward_sim f B A \<equiv> (\<forall>s. f s \<noteq> {}) \<and> (\<forall>s\<in>start B. f s \<subseteq> start A) \<and> (\<forall>s t a t'. t' \<in> f t \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> t \<and> reachable B s \<longrightarrow> (\<exists>e. fst e \<in> f s \<and> last_state e = t' \<and> is_exec_frag_of A e \<and> (let tr = trace (ioa.asig A) e in if a \<in> externals (ioa.asig A) then tr = [a] else tr = [])))"
space_F: fixes i :: "'t" shows "space (F i) = \<Omega>"
K: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" shows "H |- p -> q -> p"
edge_is_path: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q" shows "path (p ; q\<^sup>T)"
vars0_def: shows "st1.vars0 = {Suc k} \<union> {Suc (Suc k)..<n}"
A_def: fixes a :: "'b"   and t :: "'b tree" shows "S34.A a t = real (T_splay a t) + S34.\<Phi> (splay a t) - S34.\<Phi> t"
srewrite_stepI0: fixes name :: "'a"   and rhs :: "'b"   and rs :: "('a \<times> 'b) list" assumes "(name, rhs) \<in> set rs"   and "is_map (set rs)" shows "srewrite_step rs name rhs"
cod_dom: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.cod (CC.dom f) = CC.dom f"
unambigous_prefix_routing_weak_mono: fixes rr :: "('b, 'c) routing_rule_scheme"   and rtbl :: "('b, 'c) routing_rule_scheme list"   and rr' :: "('b, 'c) routing_rule_scheme" assumes "is_longest_prefix_routing (rr # rtbl)"   and "rr' \<in> set rtbl" shows "routing_rule_sort_key rr \<le> routing_rule_sort_key rr'"
stirling_sum_def: fixes j :: "nat"   and m :: "nat"   and x :: "'a" shows "stirling_sum j m x = (- (1::'a)) ^ j * (\<Sum>k = 1..<m. of_real (bernoulli (Suc k)) * pochhammer (of_nat k) j / (of_nat k * of_nat (Suc k)) * inverse x ^ (k + j))"
finite_imp_ex_se_succ: fixes c :: "('d, 'e) conf"   and l :: "('d, 'e) label" assumes "finite (pred c)"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)" shows "\<exists>c'. se c l c'"
induced_subgraph_refl: shows "induced_subgraph (G \<restriction> S) (G \<restriction> S)"
Constrains_weaken: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A Co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B Co B'"
jordancurve_1: shows "inside (path_image (p +++ linepath a b)) \<inter> outside (path_image (p +++ linepath a b)) = {}"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "nat"   and a3 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "a1 \<leadsto>\<^bsup>a2\<^esup> a3" obtains x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = xa__" and "a2 = 0" and "a3 = ya__" and "xa__ = ya__"   | x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and n :: "nat" and z :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = x" and "a2 = Suc n" and "a3 = z" and "x \<leadsto> y" and "y \<leadsto>\<^bsup>n\<^esup> z"
prefixE: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<le> ys" obtains zs :: "'c list" where "ys = xs @ zs"
subformulas\<^sub>\<mu>_list_length: fixes \<phi> :: "'b ltln" shows "length (subformulas\<^sub>\<mu>_list \<phi>) = card (subformulas\<^sub>\<mu> \<phi>)"
when_add_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b"   and P :: "bool" assumes "class.monoid_add plus zero" shows "zero.when zero (plus a b) P = plus (zero.when zero a P) (zero.when zero b P)"
member_conv_keys: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.member t = (\<lambda>x. x \<in> set (RBT_Set2.keys t))"
assms_1: shows "r \<in> carrier R - {\<zero>}"
vsv_vimage_eqI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
less_dag_set_of: fixes x :: "dag"   and y :: "dag" assumes "x < y" shows "set_of x \<subseteq> set_of y"
prod_pos': shows "0 < (\<Prod>x\<in>{0..<m} - {i}. real_of_int (d fs' x))"
numeral_plus_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m + n)"
elims_2: fixes x :: "'a list" assumes "strict_sorted x" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "Ball (set ys) ((<) xa) \<and> strict_sorted ys"
iwlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "iwlsFSwSb MOD \<equiv> iwlsFSw MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igSubstCls MOD"
scale_scale: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and x :: "'d" assumes "module scale" shows "scale a (scale b x) = scale (a * b) x"
simps_5: fixes f1 :: "'a"   and f2 :: "color \<Rightarrow> ('b, 'c) rbt \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('b, 'c) rbt \<Rightarrow> 'a"   and x21 :: "color"   and x22 :: "('b, 'c) rbt"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "('b, 'c) rbt" shows "(case Branch x21 x22 x23 x24 x25 of Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = f2 x21 x22 x23 x24 x25"
Ax: shows "A $ x $ (LEAST k. A $ x $ k \<noteq> (0::'a)) = (1::'a)"
eln_mono2: fixes x :: "ennreal"   and y :: "ennreal" assumes "x \<le> y" shows "eln x \<le> eln y"
ACI_norm_Plus: fixes r :: "'b rexp"   and s :: "'b rexp"   and t :: "'b rexp" assumes "\<guillemotleft>r\<guillemotright> = Plus s t" shows "\<exists>s t. r = Plus s t"
SIod_le: fixes T :: "'c set"   and D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<preceq>\<^bsub>SIod D T\<^esub> b = a \<preceq>\<^bsub>D\<^esub> b"
A: assumes "\<And>ps'. ps' \<in> set (split_sign_changes ps x\<^sub>0) \<Longrightarrow> \<forall>\<^sub>F x in at x\<^sub>0. sign_changes ps' x = sign_changes ps' x\<^sub>0" shows "\<forall>\<^sub>F x in at x\<^sub>0. (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x) = (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x\<^sub>0)"
Units_cong: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units R"   and "a \<sim> b"   and "b \<in> carrier R" shows "b \<in> Units R"
card_inverse: fixes R :: "('b \<times> 'a) set" shows "card (R\<inverse>) = card R"
intros_3: fixes v :: "'c"   and w :: "'c"   and vs :: "'c list" assumes "v \<rightarrow>\<^bsub>T\<^esub> w"   and "T.walk (w # vs)" shows "T.walk (v # w # vs)"
listrel_iff_nth: fixes xs :: "'c list"   and ys :: "'d list"   and r :: "('c \<times> 'd) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>n<length xs. (xs ! n, ys ! n) \<in> r))"
sfunc_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "sfunc s i j = s.funcs s ! sfunc_ind s i j"
loxodromic_imp_not_elliptic_parabolic_1: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "\<not> elliptic_isometry f"
Cod_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Cod (Inv t) = Dom t"
assms_4: shows "b \<bullet> v mod a = b \<bullet> w mod a"
disc_eq_case_2: fixes sum :: "'c + 'd" shows "(\<not> isl sum) = (case sum of Inl uu_ \<Rightarrow> False | Inr uu_ \<Rightarrow> True)"
le_\<I>_def: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('i, 'j) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
strong_induct: fixes P :: "'a \<Rightarrow> binding \<Rightarrow> bool"   and z :: "'a"   and binding :: "binding" assumes "\<And>vrs ty z. P z (VarB vrs ty)"   and "\<And>tyvrs ty z. P z (TVarB tyvrs ty)" shows "P z binding"
INF_union: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and M :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set"   and B :: "'c::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) ((M::'c::type \<Rightarrow> 'b::type) ` ((A::'c::type set) \<union> (B::'c::type set))) = (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (Inf (M ` A)) (Inf (M ` B))"
THE_defaultI': fixes P :: "'a \<Rightarrow> bool"   and d :: "'a" assumes "\<exists>!x. P x" shows "P (THE_default d P)"
abelian_relcycle_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "comm_group (relcycle_group p X S)"
hid_is_R: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "hid P cs is R healthy"
H: shows "(p\<^sub>1 = Infinity \<or> p\<^sub>2 = Infinity \<or> p\<^sub>3 = Infinity) \<or> (p\<^sub>1 = opp p\<^sub>2 \<or> p\<^sub>2 = opp p\<^sub>3) \<or> opp p\<^sub>1 = ell_field_class.add a p\<^sub>2 p\<^sub>3 \<or> opp p\<^sub>3 = ell_field_class.add a p\<^sub>1 p\<^sub>2"
add_scale_img: shows "(\<lambda>x. a + (b - a) * x) ` {0..1} = {a..b}"
simps_1: fixes x :: "'c option" shows "sup_option x None = x"
case_1: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Private of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f1"
elims_2: fixes init_s :: "(nat \<times> linear_poly) list \<Rightarrow> 'a"   and assert_s :: "'b \<times> 'c atom \<Rightarrow> 'a \<Rightarrow> 'b list + 'a"   and check_s :: "'a \<Rightarrow> 'b list + 'a"   and solution_s :: "'a \<Rightarrow> (nat, 'c) mapping"   and checkpoint_s :: "'a \<Rightarrow> 'd"   and backtrack_s :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and invariant_s :: "(nat \<times> linear_poly) list \<Rightarrow> ('b \<times> 'c atom) set \<Rightarrow> 'a \<Rightarrow> bool"   and checked_s :: "(nat \<times> linear_poly) list \<Rightarrow> ('b \<times> 'c atom) set \<Rightarrow> 'a \<Rightarrow> bool"   and preprocess :: "('b \<times> 'c ns_constraint) list \<Rightarrow> (nat \<times> linear_poly) list \<times> ('b \<times> 'c atom) list \<times> ((nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping) \<times> 'b list"   and x :: "('b \<times> 'c ns_constraint) list"   and xa :: "'b set"   and xb :: "(('b, ('b \<times> 'c atom) list) mapping \<times> ((nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping) \<times> 'b list) \<times> 'a" assumes "Incremental_Atom_Ops_For_NS_Constraint_Ops init_s assert_s check_s solution_s checkpoint_s backtrack_s invariant_s checked_s preprocess"   and "Incremental_Atom_Ops_For_NS_Constraint_Ops.invariant_nsc invariant_s preprocess x xa xb" obtains nsc :: "('b \<times> 'c ns_constraint) list" and J :: "'b set" and asi :: "('b, ('b \<times> 'c atom) list) mapping" and tv :: "(nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping" and ui :: "'b list" and s :: "'a" where "x = nsc" and "xa = J" and "xb = ((asi, tv, ui), s)" and "case preprocess nsc of (t, as, tv', ui') \<Rightarrow> Incremental_Atom_Ops_For_NS_Constraint_Ops_Default.invariant_as_asi (set as) asi tv tv' ui ui' \<and> invariant_s t (set as \<inter> J \<times> UNIV) s \<and> J \<inter> set ui = {}"
Quotient_proj_space: shows "Quotient proj_rel Proj rep_proj cr_proj_space"
leq: shows "length ts\<^sub>s\<^sub>b = length ts"
affine_dependent_def: fixes s :: "'b set" shows "affine_dependent s = (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
Ex2_def: shows "Ex2 = CONJ ` CofGroups.Ex1"
interaction_bounded_by_right_gpv: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and consider' :: "'d + 'a \<Rightarrow> bool" assumes "interaction_bounded_by consider gpv n"   and "\<And>x. consider' (Inr x) \<Longrightarrow> consider x" shows "interaction_bounded_by consider' (right_gpv gpv) n"
less_le_trans: fixes x :: "('c, 'd) Term.term"   and y :: "('c, 'd) Term.term"   and z :: "('c, 'd) Term.term" assumes "x \<sqsubset> y"   and "y \<sqsubseteq> z" shows "x \<sqsubset> z"
Not_eqvt: fixes p :: "perm"   and x :: "('e, 'f, 'g) formula" shows "p \<bullet> Formula.Not x = Formula.Not (p \<bullet> x)"
content_const: fixes c :: "'b" shows "content [:c:] = normalize c"
matrix_same: shows "M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) = kronecker_product (get_matrix (calculate_data p qs1)) (get_matrix (calculate_data p qs2))"
Cons_3: fixes Ks :: "'a list" assumes "set Ks \<subseteq> K"   and "combine Ks (u # Us) = \<zero>" shows "set (take (length (u # Us)) Ks) \<subseteq> {\<zero>}"
simps: fixes s1 :: "'e"   and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" shows "P18 (s1, s2) = (abortVals s2 \<noteq> {} \<longrightarrow> (\<exists>p. status s2 p \<noteq> Sleep))"
mem_interior_cball: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> interior S) = (\<exists>e>0. cball x e \<subseteq> S)"
valof_skip2: fixes e :: "exp"   and s :: "loc \<Rightarrow> nat"   and v :: "nat"   and s' :: "loc \<Rightarrow> nat" assumes "(e, s) -|-> (v, s')" shows "(VALOF SKIP RESULTIS e, s) -|-> (v, s')"
geodesic_segment_param_in_geodesic_spaces_4: fixes x :: "'a"   and y :: "'a" shows "isometry_on {0..dist x y} (geodesic_segment_param {x--y} x)"
subint1D: fixes I :: "qtname"   and J :: "qtname"   and G :: "prog" assumes "(I, J) \<in> subint1 G" shows "? i:iface G I: J \<in> set (isuperIfs i)"
extend_indets_inject: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "extend_indets p = extend_indets q" shows "p = q"
Inf_atLeast: fixes x :: "'a" shows "Inf {x..} = x"
Class_in_Partition: fixes a :: "'a" assumes "a \<in> S" shows "Class a \<in> Partition"
InvariantImpliedLiteralsAndElementsEntailLiteralThenDecisionsEntailLiteral: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and literal :: "Literal" assumes "InvariantImpliedLiterals F M"   and "formulaEntailsLiteral (F @ val2form (elements M)) literal" shows "formulaEntailsLiteral (F @ val2form (decisions M)) literal"
Domain_iff: fixes a :: "'l"   and r :: "('l \<times> 'm) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
sel_2: fixes x2 :: "enat" shows "epred (eSuc x2) = x2"
r: fixes x :: "'a" assumes "x \<in> S" shows "r x = x"
igSubstIGVar2STR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGVar2STR MOD \<equiv> (\<forall>ys y y1. igSubst MOD ys (igVar MOD ys y1) y (igVar MOD ys y) = igVar MOD ys y1) \<and> (\<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y)"
b: shows "b < (LEAST n. A $ i $ n \<noteq> (0::'b))"
modif_clause: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call q (Modif \<sigma>),{}"
pure_eq_iff: fixes h :: "'a"   and f :: "('a, 'c, 'b) prog" shows "(\<forall>h' x. h \<turnstile> f \<rightarrow>\<^sub>r x \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h' \<longrightarrow> h = h') = pure f h"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>* (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = fa__" and "a3 = ms__" and "a4 = s__" and "a5 = []" and "a6 = ms__" and "a7 = s__" and "length ms__ = length s__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as' :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = last as # as'" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" and "S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s'')"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "c_assoc_have_key x xa = y" obtains y :: "nat" and x :: "nat" where "xa__ = y" and "xa = x" and "ya__ = (if y = 0 then 1 else if c_fst (c_hd y) = x then 0 else c_assoc_have_key (c_tl y) x)"
Suc: shows "{i} \<times> below (2 * n) \<in> tiling domino"
S_inductive_sup_bound: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X" shows "\<forall>x\<in>X. x \<preceq> Algebra2.Sup D X"
exits_append: fixes xs :: "instr list"   and ys :: "instr list" shows "exits (xs @ ys) = exits xs \<union> (+) (size xs) ` exits ys - {0..<size xs + size ys}"
pid_label_eq: fixes pgc :: "('c, 'ls, 'gs) pid_global_config" shows "label (pidgc_\<alpha> pgc) = pid_label pgc"
coeff_monom_mult: fixes m' :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and q :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (MPoly_Type.monom m' a * q) (m' + m) = a * MPoly_Type.coeff q m"
queue_invar_remove1: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.queue_invar q" shows "BinomialHeapStruc.queue_invar (remove1 t q)"