hcju/isabelleps huggingface dataset:
key_compare_key_order_of_le: shows "key_compare key_order_of_le = (\<lambda>x y. if x < y then Lt else if x = y then Eq else Gt)"
sel_3: fixes x21 :: "'d"   and x22 :: "'d list" shows "tl (x21 # x22) = x22"
that: fixes V :: "'b set" assumes "openin (top_of_set T) V"   and "a \<in> V"   and "S \<times> V \<subseteq> U" shows "thesis"
fds_abs_converges_diff: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_abs_converges f s"   and "fds_abs_converges g s" shows "fds_abs_converges (f - g) s"
scalar_product_def: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec"   and w :: "('a, 'b) Finite_Cartesian_Product.vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $h i * w $h i)"
InvariantConsistentAfterUnitPropagate: fixes M :: "(Literal \<times> bool) list"   and uClause :: "Literal list"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantConsistent M"   and "isUnitClause uClause uLiteral (elements M)"   and "M' = M @ [(uLiteral, False)]" shows "InvariantConsistent M'"
not_int_rec: fixes k :: "int" shows "NOT k = of_bool (even k) + 2 * NOT (k div 2)"
list_of_eucl_nth: fixes n :: "nat"   and x :: "'e" assumes "n < DIM('e)" shows "list_of_eucl x ! n = x \<bullet> Basis_list ! n"
map_of_map_to_set: fixes l :: "('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst l)" shows "(map_of l = m) = (set l = map_to_set m)"
outer_pasch: fixes A :: "'p"   and C :: "'p"   and P :: "'p"   and B :: "'p"   and Q :: "'p" assumes "Bet A C P"   and "Bet B Q C" shows "\<exists>X. Bet A X B \<and> Bet P Q X"
app_vconverseI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
send_insert_id_1: fixes i :: "'a \<times> nat"   and n :: "nat"   and \<sigma> :: "'b"   and m :: "('a \<times> nat, 'b) insert_message" assumes "state i \<bind> (\<lambda>s. create_insert s n \<sigma> i) = Inr (Insert m)" shows "I m = i"
map_fun_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'd" shows "map_fun f g h = g \<circ> h \<circ> f"
hom_pairwise: fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and G :: "('c, 'f) monoid_scheme"   and H :: "('d, 'g) monoid_scheme"   and K :: "('e, 'h) monoid_scheme" shows "(f \<in> hom G (H \<times>\<times> K)) = (fst \<circ> f \<in> hom G H \<and> snd \<circ> f \<in> hom G K)"
proj2_Col_coincide: fixes a :: "proj2"   and c :: "proj2" shows "proj2_Col a a c"
infinite: fixes A :: "'b set" assumes "infinite A" shows "mset_set A = {#}"
subgroup_singular_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "subgroup (singular_relboundary_set p X S) (chain_group p X)"
scale_right_diff_distrib: fixes a :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "a *s (x - y) = a *s x - a *s y"
homeomorphism_ident: fixes T :: "'b set" shows "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
degree_aform_err_cos_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "cos_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> Partition" shows "a \<in> M"
MGF_asm: fixes A :: "'a set"   and t :: "'c" assumes "\<forall>pn\<in>U. P A {mgf_call pn}" shows "P A {mgf t}"
normal_moment_even: shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k)) (fact (2 * k) / ((2 / \<sigma>\<^sup>2) ^ k * fact k))"
simps_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f x [] = [x]"
prj_simps_1: shows "C.arr \<mu>\<nu>.prj\<^sub>1"
nonneg_mat_def: fixes A :: "'a mat" shows "nonneg_mat A \<equiv> \<forall>a\<in>elements_mat A. (0::'a) \<le> a"
apfst_conv: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'d"   and y :: "'c" shows "apfst f (x, y) = (f x, y)"
that: fixes CAs0 :: "'a literal multiset list"   and \<eta>s0 :: "'s list"   and DA0 :: "'a literal multiset"   and \<eta>0 :: "'s"   and As0 :: "'a list"   and AAs0 :: "'a multiset list"   and D0 :: "'a literal multiset"   and Cs0 :: "'a literal multiset list" assumes "length CAs0 = n"   and "length \<eta>s0 = n"   and "DA0 \<in> M"   and "DA0 \<cdot> \<eta>0 = DA"   and "S DA0 \<cdot> \<eta>0 = S_M S M DA"   and "\<forall>CA0\<in>set CAs0. CA0 \<in> M"   and "CAs0 \<cdot>\<cdot>cl \<eta>s0 = CAs"   and "map S CAs0 \<cdot>\<cdot>cl \<eta>s0 = map (S_M S M) CAs"   and "is_ground_subst \<eta>0"   and "is_ground_subst_list \<eta>s0"   and "As0 \<cdot>al \<eta>0 = As"   and "AAs0 \<cdot>\<cdot>aml \<eta>s0 = AAs"   and "length As0 = n"   and "D0 \<cdot> \<eta>0 = D"   and "DA0 = D0 + negs (mset As0)"   and "S_M S M (D + negs (mset As)) \<noteq> {#} \<Longrightarrow> negs (mset As0) = S DA0"   and "length Cs0 = n"   and "Cs0 \<cdot>\<cdot>cl \<eta>s0 = Cs"   and "\<forall>i<n. CAs0 ! i = Cs0 ! i + poss (AAs0 ! i)"   and "length AAs0 = n" shows "thesis"
atLeast0_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {0..<Suc n} = g 0 + sum (g \<circ> Suc) {0..<n}"
nodes_empty_iff_empty_2: fixes G :: "'b ugraph" shows "({} = nodes G) = (G = graph_empty)"
StrongEq_sym: fixes X :: "'b state \<times> 'b state \<Rightarrow> 'c"   and Y :: "'b state \<times> 'b state \<Rightarrow> 'c" shows "(X \<triangleq> Y) = (Y \<triangleq> X)"
simps_1: fixes x5 :: "char list"   and y5 :: "char list" shows "(Class x5 = Class y5) = (x5 = y5)"
e_2: fixes x :: "float"   and y :: "float" assumes "x \<in> U"   and "dist (real_of_float y) (real_of_float x) < e x" shows "y \<in> U"
supp_rel_tNot: fixes t :: "('a, 'b, 'c, 'd) Tree" shows "supp_rel (=\<^sub>\<alpha>) (tNot t) = supp_rel (=\<^sub>\<alpha>) t"
acyclic_5e_def: fixes x :: "'a" shows "acyclic_5e x \<equiv> \<forall>y z. y \<le> x \<and> z \<le> x \<and> y \<sqinter> z = bot \<longrightarrow> y\<^sup>\<star> \<sqinter> z\<^sup>\<star> = (1::'a)"
relax_outgoing''_def: fixes l :: "('d \<times> 'c) list"   and du :: "'d"   and V :: "'c \<Rightarrow> 'e option"   and Q :: "'c \<Rightarrow> 'd option" shows "relax_outgoing'' l du V Q = fold (\<lambda>(d, v) Q. case Q v of None \<Rightarrow> if v \<in> dom V then Q else Q(v \<mapsto> du + d) | Some d' \<Rightarrow> Q(v \<mapsto> min (du + d) d')) l Q"
af_G_letter_free_F_1: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" assumes "\<^bold>G \<phi> = {}" shows "\<^bold>G (af_letter \<phi> \<nu>) = {}"
Can_red_1: fixes a :: "'c local.term" assumes "Ide a" shows "Can (a\<^bold>\<down>)"
\<omega>_def: shows "\<omega> = cis (2 * pi / real n)"
eq_\<I>_gpvD_results_gpv2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv"   and y :: "'b" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "y \<in> results_gpv \<I> gpv'" shows "\<exists>x\<in>results_gpv \<I> gpv. A x y"
tagged_fun_relD_rhs: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'h"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and x :: "'e"   and x' :: "'g" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, Let x' f') \<in> B"
AbstAtomicP_graph_aux_def: shows "AbstAtomicP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>AbstAtomicP_aux t v i y y' t' u u'. x1 = (v, i, y, y') \<and> x2 = SyntaxN.Ex t (SyntaxN.Ex u (SyntaxN.Ex t' (SyntaxN.Ex u' (AbstTermP v i (Var t) (Var t') AND AbstTermP v i (Var u) (Var u') AND (y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u') OR y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u')))))) \<and> atom t \<sharp> (v, i, y, y', t', u, u') \<and> atom t' \<sharp> (v, i, y, y', u, u') \<and> atom u \<sharp> (v, i, y, y', u') \<and> atom u' \<sharp> (v, i, y, y'))"
INF_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "(\<Sqinter>(f ` A)) x = (\<Sqinter>y\<in>A. f y x)"
mk_minsky_seq_map: fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and g :: "nat \<Rightarrow> nat"   and R :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "mk_minsky P"   and "mk_minsky Q"   and "inj g"   and "\<And>vs vs' vs''. \<lbrakk>P vs vs'; Q vs' vs''\<rbrakk> \<Longrightarrow> R vs vs''" shows "mk_minsky (\<lambda>vs vs'. R (vs \<circ> g) (vs' \<circ> g) \<and> (\<forall>x. x \<notin> range g \<longrightarrow> vs x = vs' x))"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Vertex x1 = Vertex y1) = (x1 = y1)"
compose_action_def: fixes f1 :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option"   and f2 :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option"   and x :: "predicate \<times> object list \<Rightarrow> bool" shows "compose_action f1 f2 x = (case f2 x of None \<Rightarrow> None | Some y \<Rightarrow> f1 y)"
intro: fixes m :: "int" assumes "poly_mod_type TYPE('a) m" shows "poly_mod_prime_type TYPE('a) m"
iTILL_iprev: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [\<dots>n]" shows "iprev t [\<dots>n] = t - Suc 0"
distinct_adj_append_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "distinct_adj (xs @ ys) = (distinct_adj xs \<and> distinct_adj ys \<and> (xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys))"
sup_left_top: fixes x :: "'c" shows "top \<squnion> x = top"
simps_2: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and v :: "'a" shows "pcas u (e # es) v = (fst e = u \<and> pcas (snd e) es v)"
set_shadow_root_locs_impl: shows "set_shadow_root_locs = set_shadow_root_locs"
th: shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (Polynomial_Expression.degree p))) (behead p)) = Ipoly bs p"
invertible_mat_four_block_mat_lower_right_id: fixes A :: "'a mat"   and m :: "nat"   and B :: "'a mat"   and n :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat m m"   and "B = 0\<^sub>m m (n - m)"   and "C = 0\<^sub>m (n - m) m"   and "D = 1\<^sub>m (n - m)"   and "m < n"   and "invertible_mat A" shows "invertible_mat (four_block_mat A B C D)"
Un_upper1: fixes A :: "'k set"   and B :: "'k set" shows "A \<subseteq> A \<union> B"
inverse_arrows_\<iota>: shows "inverse_arrows \<iota> \<d>[\<one>]"
b_def': shows "b = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var (Suc 0))"
ps_first: fixes ps :: "real poly list"   and p :: "real poly" assumes "sturm_seq ps p" shows "ps ! 0 = p"
hull_concat_lists0: fixes w :: "'b list"   and G :: "'b list set" assumes "w \<in> \<langle>G\<rangle>" shows "\<exists>ws\<in>lists G. concat ws = w"
is_nth_power_1: fixes n :: "nat" shows "is_nth_power n (1::'a)"
hom_closed: shows "f ` U\<^sub>M\<^sub>_\<^sub>1 \<subseteq> U\<^sub>M\<^sub>_\<^sub>2"
local_assms: shows "to_ct_list ts = (a # as) @ bs"
d_delta_lnexp_cf4: fixes x :: "real" shows "((\<lambda>x. ln (exp_cf4 x) - x) has_real_derivative diff_delta_lnexp_cf4 x) (at x)"
assms_3: shows "(f has_derivative f') (at x within cbox a b)"
bfs2_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and succ :: "nat \<Rightarrow> nat list nres"   and src :: "nat"   and dst :: "nat" shows "Graph.bfs2 c succ src dst \<equiv> if src = dst then return (Some []) else Graph.pre_bfs2 c succ src dst \<bind> (\<lambda>br. case br of None \<Rightarrow> return None | Some (d, PRED) \<Rightarrow> Graph.extract_rpath c src dst PRED \<bind> (\<lambda>p. return (Some p)))"
skelBinp_def: fixes binp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs set option" shows "skelBinp binp = qSkelBinp (pickBinp binp)"
field_is_idom: fixes K :: "('e, 'f) Ring_scheme" assumes "Corps K" shows "Idomain K"
linear_injective_imp_surjective: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "surj f"
inverse_le_imp_le: fixes a :: "'a"   and b :: "'a" assumes "inverse a \<le> inverse b"   and "(0::'a) < a" shows "b \<le> a"
FOLD_bin_ne_def: fixes b :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" shows "FOLD_bin_ne b X = ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>x. x \<in> X) \<bind> (\<lambda>x. let X = X - {x} in FOREACH X (\<lambda>x r. RETURN (b x r)) x))"
openin_interior_of: fixes X :: "'c topology"   and S :: "'c set" shows "openin X (X interior_of S)"
Int_Un_eq_4: fixes T :: "'k set"   and S :: "'k set" shows "T \<union> S \<inter> T = T"
V_1: fixes j :: "nat" shows "closure (V j) \<subseteq> V (Suc j)"
extremum_unique: fixes a :: "'a" shows "(\<top> \<le> a) = (a = \<top>)"
blue_leaves: fixes w :: "'person \<Rightarrow> 'color"   and p :: "'person"   and n :: "nat" assumes "w p = blue"   and "local.valid w"   and "w guru \<noteq> blue" shows "local.leaves n p w = (card (local.blues_seen w p) \<le> n)"
bezout_matrix_works2: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'e"   and b :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and j :: "'f" assumes "is_bezout_ext bezout"   and "a \<noteq> b"   and "A $ a $ j \<noteq> (0::'d) \<or> A $ b $ j \<noteq> (0::'d)" shows "(bezout_matrix A a b j bezout ** A) $ b $ j = (0::'d)"
assms: shows "\<forall>PROB as s. P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
lebesgue_sets_translation: fixes S :: "'a set"   and a :: "'a" assumes "S \<in> sets lebesgue" shows "(+) a ` S \<in> sets lebesgue"
fstar_unfoldl: fixes f :: "'b \<Rightarrow> 'b" assumes "Inf_pres f" shows "id \<sqinter> (f \<circ> fstar f) = fstar f"
set_iterator_product_foldli_conv: fixes as :: "'b list"   and bs :: "'b \<Rightarrow> 'c list" shows "set_iterator_product (foldli as) (\<lambda>a. foldli (bs a)) = foldli (concat (map (\<lambda>a. map (Pair a) (bs a)) as))"
st: shows "s \<rightarrow>\<^bsub>T\<^esub> t"
simps_9: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Gt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f3"
del_UnEdge_even_even: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "finite (edges G)"   and "finite (nodes G)"   and "(v, w, v') \<in> edges G"   and "even (degree v G)"   and "even (degree v' G)" shows "num_of_odd_nodes (del_unEdge v w v' G) = num_of_odd_nodes G + 2"
PBQ: shows "(P, B, Q) = diagonal_to_Smith_PQ A bezout"
simps_2: fixes v :: "'c"   and l :: "'c list" shows "hs_of_list (v # l) = HeapFunctional.insert v (hs_of_list l)"
divide: fixes g1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g2 :: "'b \<Rightarrow> 'c"   and f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g2 x \<noteq> (0::'c)"   and "f1 \<in> \<Theta>[F](f2)"   and "g2 \<in> \<Theta>[F](g1)" shows "(\<lambda>x. f1 x / g1 x) \<in> \<Theta>[F](\<lambda>x. f2 x / g2 x)"
arith_simps_16: shows "neg_numeral_class.sub num.One num.One = (0::'b)"
lang_def: fixes G :: "('a, 'b, 'c) sa_rec_scheme" assumes "sa G" shows "sa.lang G \<equiv> Collect (sa.accept G)"
unit_in_vhom: shows "\<guillemotleft>\<eta>\<^sub>0 : src e\<^sub>0 \<Rightarrow> d\<^sub>0 \<star> e\<^sub>0\<guillemotright>"
assms_2: shows "rk {A, B, C, A', B', C', P} = 3"
poly_add: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "poly (p + q) x = poly p x + poly q x"
degree_Poly_berlekamp_le: fixes i :: "nat"   and u :: "'a mod_ring poly" assumes "i < degree u" shows "degree (Poly (list_of_vec (row (local.berlekamp_mat u) i))) < degree u"
rel_fundef_generalize: fixes fd1 :: "('dyn, 'a, 'b, 'c, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 (Subx.generalize_fundef fd2)"
Smith_normal_form_mat_def: fixes A :: "'a mat" shows "Smith_normal_form_mat A = ((\<forall>a. a + 1 < min (dim_row A) (dim_col A) \<longrightarrow> A $$ (a, a) dvd A $$ (a + 1, a + 1)) \<and> isDiagonal_mat A)"
to_contain_is_to_divide: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "(PIdl b \<subseteq> PIdl a) = a divides b"
Keys_Macaulay_red: fixes fs :: "('t \<Rightarrow>\<^sub>0 'd) list" shows "Keys (set (Macaulay_red (Keys_to_list fs) fs)) \<subseteq> Keys (set fs)"
CastNull\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
inj: shows "inj_on (insert n) {K. K \<subseteq> {0..<n} \<and> card K = k}"
\<C>_vars_of_\<C>_vars_empty: fixes x :: "'Var"   and y :: "'Var" assumes "x \<in> \<C>_vars y" shows "\<C>_vars x = {}"
hyps_3: shows "\<tau> # \<tau>s = strip_all_vars B"
left: assumes "\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)" shows "(\<exists>x. eval (elimVar var L F (Eq p)) (xs @ x # \<Gamma>)) \<or> (\<forall>x. aEval (Eq p) (xs @ x # \<Gamma>))"
Union_disjoint: fixes C :: "'g set set"   and A :: "'g set" shows "(\<Union> C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
sh_ok: shows "P,h \<turnstile>\<^sub>s sh \<surd>"
pw_eq_on_subset: fixes S :: "'b set"   and S' :: "'b set"   and w :: "'c \<Rightarrow> 'b set"   and w' :: "'c \<Rightarrow> 'b set" assumes "S \<subseteq> S'"   and "pw_eq_on S' w w'" shows "pw_eq_on S w w'"
le_neq_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
maximalidealI: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R"   and "carrier R \<noteq> I"   and "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R" shows "maximalideal I R"
C1_differentiable_on_const: fixes a :: "'b"   and S :: "real set" shows "(\<lambda>z. a) C1_differentiable_on S"
index_mat_four_block_3: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" shows "dim_col (four_block_mat A B C D) = dim_col A + dim_col D"
in_set_real_plus_interval: fixes x :: "real"   and X :: "float interval"   and y :: "real"   and Y :: "float interval" assumes "x \<in>\<^sub>r X"   and "y \<in>\<^sub>r Y" shows "x + y \<in>\<^sub>r X + Y"
closest_point_in_rel_interior: fixes S :: "'a set"   and x :: "'a" assumes "closed S"   and "S \<noteq> {}"   and "x \<in> affine hull S" shows "(closest_point S x \<in> rel_interior S) = (x \<in> rel_interior S)"
singleton_equal_aux: fixes p :: "'a"   and q :: "'a" assumes "singleton p"   and "singleton q"   and "q \<le> p" shows "p \<le> q ; 1"
circuit_iff_rank_of: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "\<E>.circuit X = (X \<noteq> {} \<and> (\<forall>x\<in>X. \<E>.rank_of (X - {x}) = card (X - {x}) \<and> card (X - {x}) = \<E>.rank_of X))"
LabelFun_LabelFunSem: fixes a :: "('d, 'e, 'f) hierauto" shows "LabelFun (HA2Kripke a) = LabelFunSem a"
induct: fixes P :: "'b rexp \<Rightarrow> bool"   and rexp :: "'b rexp" assumes "P Zero"   and "P One"   and "\<And>x. P (Atom x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Plus x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Times x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (Star x)" shows "P rexp"
assms_3: shows "Domain f_a \<inter> Domain f_b = Domain (f_a \<inter> f_b)"
hyps_1: fixes gpv :: "('a, 'out, 'in) gpv" shows "interaction_bound' gpv \<le> interaction_bound consider gpv"
le_supE: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b"   and x :: "'a \<times> 'b" assumes "less_eq_prod (sup_prod a b) x" obtains "less_eq_prod a x" and "less_eq_prod b x"
elims_2: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and x :: "v sem_env" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_all_env static_cenv x" obtains v0 :: "(char list \<times> v) list" and c0 :: "(char list, char list, nat \<times> tid_or_exn) namespace" where "x = make_sem_env (Bind v0 []) c0" and "c0 = static_cenv \<and> list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) v0"
inv_mult_group: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = \<ominus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> x"
formulaEntailed: shows "\<forall>clause. clause el formula \<longrightarrow> formulaEntailsClause formula0 clause"
nn_integral_dominated_convergence: fixes u :: "nat \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure"   and u' :: "'c \<Rightarrow> ennreal"   and w :: "'c \<Rightarrow> ennreal" assumes "\<And>i. u i \<in> borel_measurable M"   and "u' \<in> borel_measurable M"   and "w \<in> borel_measurable M"   and "\<And>j. AE x in M. u j x \<le> w x"   and "integral\<^sup>N M w < \<infinity>"   and "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x" shows "(\<lambda>i. integral\<^sup>N M (u i)) \<longlonglongrightarrow> integral\<^sup>N M u'"
bdd_above_Ioo: fixes a :: "'a"   and b :: "'a" shows "bdd_above {a<..<b}"
elims_2: fixes x :: "('c, 'd, 'e, 'f) edge_kind"   and xa :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "local.CFG.pred x xa" obtains f :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = \<Up>f__" and "xa = cf__ # cfs__"   | Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "Q__ (fst cfa__)"   | 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 cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "Qa__ (fst cfb__, r)"   | Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and p :: "'f" and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "Q cf \<and> cfs \<noteq> []"
nt_sum: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "sfg A a" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
abs_eq: shows "0 = Blinfun (\<lambda>x. 0::'b)"
diff_add_assoc: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "c + b - a = c + (b - a)"
adjunct_ord_large_a: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "Order D"   and "a \<notin> carrier D" shows "\<forall>x\<in>carrier D. x \<prec>\<^bsub>adjunct_ord D a\<^esub> a"
mod_exp_nat_def: fixes b :: "nat"   and e :: "nat"   and m :: "nat" shows "mod_exp_nat b e m = b ^ e mod m"