hcju/isabelleps huggingface dataset:
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset" assumes "design point_set block_collection"   and "resolvable_design_axioms point_set block_collection partition" shows "resolvable_design point_set block_collection partition"
le_fin_range: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2" shows "set w\<^sub>1 \<subseteq> set w\<^sub>2"
apply_cltn2_line_injective: fixes l :: "proj2_line"   and C :: "cltn2"   and m :: "proj2_line" assumes "apply_cltn2_line l C = apply_cltn2_line m C" shows "l = m"
cases: fixes a :: "'outPort"   and n :: "'node"   and p :: "'inPort" assumes "a \<in> hyps_for' n p" obtains h :: "'outPort" where "a = h" and "hyps n h = Some p"
emp_any_per: fixes r :: "'b list" assumes "r \<noteq> \<epsilon>" shows "\<epsilon> \<le>p r\<^sup>\<omega>"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'b set" assumes "n \<le> card S" obtains T :: "'b set" where "T \<subseteq> S" and "card T = n" and "finite T"
rewriteL_comp_comp2: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and l1 :: "'e \<Rightarrow> 'c"   and l2 :: "'b \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'b"   and r :: "'f \<Rightarrow> 'e" assumes "f \<circ> g = l1 \<circ> l2"   and "l2 \<circ> h = r" shows "f \<circ> (g \<circ> h) = l1 \<circ> r"
nt_abbrev_def: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and F :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and G :: "('d, 'e, 'f, 'g, 'h, 'i) Functor" shows "NT f : F \<Longrightarrow> G \<equiv> NatTrans f \<and> NTDom f = F \<and> NTCod f = G"
seq_incr_Suc: fixes i :: "nat" shows "seq i \<subseteq> seq (Suc i)"
gcd_abs1_int: fixes x :: "int"   and y :: "int" shows "gcd \<bar>x\<bar> y = gcd x y"
ide_iso_in_hom: fixes a :: "'a \<times> 'a" assumes "VoV.ide a" shows "VoV.iso_in_hom a a a"
Compl_set_less_aux_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and proper_interval :: "'c option \<Rightarrow> 'c option \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and ao :: "'c option"   and s1 :: "'d"   and s2 :: "'e" shows "proper_intrvl.Compl_set_less_aux_fusion less proper_interval g1 g2 ao s1 s2 = proper_intrvl.Compl_set_less_aux less proper_interval ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
OK_check: fixes eX :: "'c withERR" assumes "eX \<noteq> ERR" shows "OK (check eX) = eX"
is_rbt_rbtreeify: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and kvs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map fst kvs)"   and "distinct (map fst kvs)" shows "ord.is_rbt less (rbtreeify kvs)"
hyps_2: shows "t \<in> value \<or> Ex ((\<longmapsto>) t)"
G: shows "filter (\<lambda>x. key x = k) (nths xs A) = y # filter (\<lambda>x. key x = k) (nths vs {i. Suc i + mini xs key \<in> A})"
convex_hull_subset_affine_hull: fixes s :: "'b set" shows "convex hull s \<subseteq> affine hull s"
prems_2: shows "\<forall>a\<in>insert x J. i dvd a"
simps_1: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" shows "pi \<bullet> Some x = Some (pi \<bullet> x)"
weak_tau_modality_eq_iff: fixes x :: "('idx, 'pred, 'act) formula"   and y :: "('idx, 'pred, 'act) formula" shows "(weak_tau_modality x = weak_tau_modality y) = (x = y)"
id_comp: fixes g :: "'d \<Rightarrow> 'e" shows "id \<circ> g = g"
final_finished: fixes s :: "('fenv, 'henv, ('fun, 'dyn) frame) state" assumes "final s" shows "finished (\<rightarrow>) s"
ltl_implies_satisfiable_prefix: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and i :: "nat" assumes "w \<Turnstile>\<^sub>n \<phi>" shows "\<not> af \<phi> (prefix i w) \<sim>\<^sub>L false\<^sub>n"
R3: fixes x :: "'e"   and y :: "'f" assumes "T3 x y" shows "R3 x (Rep3 y)"
p: shows "poly_mult_rat r1 p2 \<noteq> 0"
Domain_iff: fixes a :: "'b"   and r :: "('b \<times> 'd) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
assms_4: shows "c_k i (j + 1) k' \<le> c_k i (j + 1) k"
cases: fixes a :: "('Var, 'AExp, 'BExp) Stmt" assumes "is_final a" obtains "a = Stop"   | c :: "('Var, 'AExp, 'BExp) Stmt" and m :: "'Var ModeUpd" where "a = c@[m]" and "is_final c"
sup1E: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "(A \<squnion> B) x" obtains "A x"   | "B x"
infs_mono: fixes w :: "'b stream"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>a. \<lbrakk>a \<in> sset w; P a\<rbrakk> \<Longrightarrow> Q a"   and "infs P w" shows "infs Q w"
Min_eqI: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set"   and x :: "'g" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq x y"   and "x \<in> A" shows "linorder.Min less_eq A = x"
simplify_pathD: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v" shows "\<exists>p'. path g u p' v \<and> simple p' \<and> set p' \<subseteq> set p"
dom_def: fixes m :: "'b \<Rightarrow> 'c option" shows "dom m = {a. m a \<noteq> None}"
leq_prod_def: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" shows "p \<sqsubseteq> q \<equiv> fst p \<sqsubseteq> fst q \<and> snd p \<sqsubseteq> snd q"
singleton: fixes x :: "complex" shows "complex_lex.Max {x} = x"
lt: shows "lt_vec (sr *s pow_A_1 u) (A *v pow_A_1 u)"
inline_def: fixes callee :: "'e \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'e, 'f, 'g) gpv"   and c :: "('d, 'h, 'i) gpv"   and s :: "'e" shows "inline callee c s = inline_aux callee (Inl (c, s))"
evaluation_map_axioms: shows "evaluation_map (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
mat_inverse_def: fixes A :: "'a mat" shows "mat_inverse A = (if dim_row A = dim_col A then let one = 1\<^sub>m (dim_row A) in case gauss_jordan A one of (B, C) \<Rightarrow> if B = one then Some C else None else None)"
last_appendL: fixes ys :: "'b list"   and xs :: "'b list" assumes "ys = []" shows "last (xs @ ys) = last xs"
matrix_mult_transpose_dot_column: fixes A :: "((real, 'a) vec, 'b) vec" shows "Finite_Cartesian_Product.transpose A ** A = (\<chi>i j. column i A \<bullet> column j A)"
continuous_on_scaleR: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"
simple_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf" assumes "Finite_Markov_Decision_Process K S" shows "Finite_Markov_Decision_Process.simple K S ct = memoryless_on (\<lambda>s. if s \<in> S then ct s else Markov_Decision_Process.arb_act K s)"
real_distribution_def: fixes M :: "real measure" shows "real_distribution M \<equiv> prob_space M \<and> real_distribution_axioms M"
th2: shows "(\<chi>k. if k = i then row i A else row k A) = A"
iT_Minus_def: fixes k :: "nat"   and I :: "nat set" shows "k \<ominus> I \<equiv> {x. x \<le> k \<and> k - x \<in> I}"
inv_comp_left_2: fixes g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" assumes "S.iso (g \<cdot>\<^sub>S f)"   and "S.iso g" shows "S.iso f"
abs_eq: fixes xa :: "int"   and x :: "int" shows "integer_of_int xa mod integer_of_int x = integer_of_int (xa mod x)"
R2_proj_R1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and i :: "nat" assumes "\<psi> \<in> \<R>\<^sup>2" shows "\<psi> i \<in> \<R>"
valid_countings: fixes a :: "nat"   and b :: "nat" shows "(a + b) * valid_countings a b = (a - b) * (a + b choose a)"
cdi_path_swap_le: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>'"   and "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> =\<^bsub>n\<^esub> \<pi>'"   and "j \<le> n" shows "j cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"
tcopy_loop_def: shows "tcopy_loop \<equiv> [(R, 0), (R, 2), (R, 3), (W0, 2), (R, 3), (R, 4), (W1, 5), (R, 4), (L, 6), (L, 5), (L, 6), (L, 1)]"
pmf_return: fixes x :: "'c"   and y :: "'c" shows "pmf (return_pmf x) y = indicat_real {y} x"
sinpoly_Suc: fixes n :: "nat" shows "sinpoly (Suc n) = (\<lambda>x. sinpoly n x + sin_coeff n * x ^ n)"
WhileTrue: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c :: "com"   and cs :: "com list" assumes "b s" shows "((WHILE b DO c) # cs, s) \<rightarrow> (c # (WHILE b DO c) # cs, s)"
cross3_simps_7: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
sum: fixes g :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "a *s sum g A = (\<Sum>x\<in>A. a *s g x)"
mult_scalar_one: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair 1 p = p"
weak_get_put: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
hom: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "continuous_on s f"   and "f ` s \<subseteq> U"   and "P f"   and "continuous_on s g"   and "g ` s \<subseteq> U"   and "P g" shows "homotopic_with_canon P s U f g"
r_kind_prim: shows "prim_recfn 1 r_kind"
CallParams: fixes es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s\<rangle> \<rightarrow> \<langle>Val v\<bullet>M(es'),s'\<rangle>"
T: shows "T \<in> FGModuleHomSet G (\<cdot>) V (\<star>) W"
recv_marker_means_cs_Done: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "t ! i = RecvMarker cid p q"   and "i < length t" shows "snd (cs (S t (i + 1)) cid) = Done"
\<beta>_3: shows "{} \<turnstile> \<alpha> IFF \<beta>"
maps_simps_1: fixes f :: "'b \<Rightarrow> 'a list"   and x :: "'b"   and xs :: "'b list" shows "List.maps f (x # xs) = f x @ List.maps f xs"
IH_2: fixes f :: "nat \<Rightarrow> real" assumes "\<And>x. x < count_weights True (remove_weights m2) \<Longrightarrow> f x = extract_weights True m2 x" shows "m2 = insert_weights True (remove_weights m2) f"
lm013: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R = R +< {x} \<times> R `` {x}"
steps'_distinct: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list" assumes "length qs = length as"   and "length as = n" shows "distinct (steps' s qs as n) = distinct s"
nb\<^sub>1\<^sub>1: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> set (effect_of op)"   and "(s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))) v \<noteq> None" shows "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op')))"
le: shows "(pp \<le> x + x') = (int_of_uint32 pp \<le> int_of_uint32 x + int_of_uint32 x')"
constraints_of_mat_vec_solution_main: fixes A :: "rat mat"   and nr :: "nat"   and nc :: "nat"   and x :: "rat vec"   and b :: "rat vec"   and c :: "rat le_constraint" assumes "A \<in> carrier_mat nr nc"   and "x \<in> carrier_vec nc"   and "b \<in> carrier_vec nr"   and "A *\<^sub>v x \<le> b"   and "c \<in> constraints_of_mat_vec A b" shows "val_of_vec x \<Turnstile>\<^sub>l\<^sub>e c"
vifintersection_vsingleton_is: fixes i :: "V"   and f :: "V \<Rightarrow> V" shows "\<Inter>\<^sub>\<circ> (VLambda (set {i}) f `\<^sub>\<circ> set {i}) = f i"
that: fixes x :: "V" assumes "a \<in>\<^sub>\<circ> x"   and "x \<in>\<^sub>\<circ> A" shows "thesis"
term_ok'_subst_typ': fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and insts :: "((variable \<times> String.literal set) \<times> typ) list" assumes "is_std_sig \<Sigma>"   and "term_ok' \<Sigma> t"   and "distinct (map fst insts)"   and "list_all (typ_ok_sig \<Sigma>) (map snd insts)" shows "term_ok' \<Sigma> (subst_typ' insts t)"
inf1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> k \<sqinter> a" shows "A \<sqinter> b \<equiv> k \<sqinter> (a \<sqinter> b)"
emeasure_distr_restrict: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and M' :: "'b measure"   and N' :: "'c measure"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> M' \<rightarrow>\<^sub>M N'"   and "A \<in> sets N'"   and "sets M' \<subseteq> sets M"   and "sets N' \<subseteq> sets N"   and "\<And>X. X \<in> sets M' \<Longrightarrow> emeasure M X = emeasure M' X"   and "\<And>X. \<lbrakk>X \<in> sets M; X \<subseteq> space M - space M'\<rbrakk> \<Longrightarrow> emeasure M X = 0" shows "emeasure (distr M N f) A = emeasure (distr M' N' f) A"
rt_strictly_fresher_def'': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "(rt1 \<sqsubset>\<^bsub>i\<^esub> rt2) = (rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2 \<and> \<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1)"
simps_1: fixes v1 :: "'d"   and v2 :: "'d" shows "Graph.ces v1 [] v2 = (v1 = v2)"
smult_one: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<one> \<odot>\<^bsub>P\<^esub> x = x"
0: shows "y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> (y mod p ^ k) = \<zero>\<^bsub>residue_ring (p ^ k)\<^esub>"
cterms_def': fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "cterms \<Gamma> = {uu_. \<exists>p pn. uu_ = p \<and> p \<in> ctermsl (\<Gamma> pn) \<and> not_call p}"
\<tau>move2While4: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and c :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (while (c) e) (Suc (Suc (Suc (length (compE2 c) + length (compE2 e))))) None"
cSup_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_above less_eq X" shows "Sup (insert a X) = sup a (Sup X)"
poly_eval_alt: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a p = (\<Sum>t\<in>keys p. lookup p t * (\<Prod>x\<in>keys t. a x ^ lookup t x))"
\<Z>_srng_length: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "vcard \<SS> = 3\<^sub>\<nat>"
0: fixes a :: "'a set"   and b :: "'a set" shows "((a, b) \<in> (set TI)\<^sup>+) = in_trancl TI a b"
set_s'_A: fixes n :: "nat"   and m :: "nat" shows "set (s'_A n m) = set init"
S_eq: fixes n :: "nat"   and x :: "real" assumes "2 \<le> n" shows "local.S (real n) (real n + x) = (ln (G (real n + x)) - ln (fact (n - 1))) / x"
RestrictedP_HFun_Sigma: fixes s :: "tm"   and k :: "tm"   and s' :: "tm" shows "{HFun_Sigma s, RestrictedP s k s'} \<turnstile> HFun_Sigma s'"
space_in_prod_algebra: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
append1_eq_conv: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and y :: "'a" shows "(append xs [x] = append ys [y]) = (xs = ys \<and> x = y)"
unit_char': fixes a :: "'b" assumes "B.obj a" shows "HoF.unit a = H (\<tau>.F.unit a) \<cdot>\<^sub>D H.unit (\<tau>.F.map\<^sub>0 a)"
PhiPlus_is_Phi_Suc: fixes n :: "nat" assumes "n < length qs" shows "\<Phi>\<^sup>+ n = \<Phi> (Suc n)"
assms: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}"
star_false_right: fixes P :: "assn" shows "P * false = false"
vintersectionD2: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> B"
converse_product_overlap: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CM"   and "O x y"   and "O z (x \<otimes> y)" shows "O z y"
elims_3: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c \<times> 'd Filters.filter"   and xa :: "'c \<times> 'd Filters.filter" assumes "triple phi"   and "\<not> triple.pairs_less_eq x xa" obtains xb :: "'c" and y :: "'d Filters.filter" and z :: "'c" and w :: "'d Filters.filter" where "x = (xb, y)" and "xa = (z, w)" and "\<not> (xb \<le> z \<and> w \<le> y)"
G: shows "Crypt (sesK (c * f)) \<lbrace>pubAK (c * (s + a * b)), Auth_Data g b, pubAK g, Crypt (priSK CA) (Hash (pubAK g))\<rbrace> \<in> synth (analz (A \<union> spies evsFR4))"
envsOfS_preserves_wls: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsPar P"   and "rho \<in> envsOfS P" shows "wlsEnv rho"
roofedI: fixes G :: "('a, 'b) graph_scheme"   and x :: "'a"   and B :: "'a set"   and S :: "'a set" assumes "\<And>p y. \<lbrakk>path G x p y; y \<in> B\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S" shows "x \<in> roofed_gen G B S"
CatIdDomCod_1: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> X) = X"
det_fold_set: fixes c' :: "'h \<Rightarrow> bool"   and f' :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and \<sigma>' :: "'h"   and result :: "'g set \<Rightarrow> 'h"   and Rk :: "('i \<times> 'g) set"   and Rs :: "('i \<times> 'g) set \<Rightarrow> ('j \<times> 'g set) set"   and tsl :: "'j \<Rightarrow> 'i list"   and s :: "'j"   and s' :: "'g set"   and c :: "'k \<Rightarrow> bool"   and R\<sigma> :: "('k \<times> 'h) set"   and f :: "'i \<Rightarrow> 'k \<Rightarrow> 'k"   and \<sigma> :: "'k" assumes "det_fold_set (\<lambda>_ _. True) c' f' \<sigma>' result"   and "is_set_to_list Rk Rs tsl"   and "(s, s') \<in> \<langle>Rk\<rangle>Rs"   and "(c, c') \<in> R\<sigma> \<rightarrow> bool_rel"   and "\<And>x y. \<lbrakk>(x, y) \<in> Rk; y \<in> s'\<rbrakk> \<Longrightarrow> (f x, f' y) \<in> R\<sigma> \<rightarrow> R\<sigma>"   and "(\<sigma>, \<sigma>') \<in> R\<sigma>" shows "(foldli (tsl s) c f \<sigma>, result s') \<in> R\<sigma>"
FinDimVectorSpaceI: assumes "\<exists>ms. set ms \<subseteq> V \<and> RSpan ms = V" shows "FinDimVectorSpace (\<cdot>) V"
simps_13: fixes p :: "fm"   and q :: "fm" shows "not (Or p q) = And (not p) (not q)"
rupture_surj_composed_with_pmod: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.rupture_surj K p q = UP.rupture_surj K p (UP.pmod q p)"
find_prime: fixes f :: "nat \<Rightarrow> bool" assumes "\<exists>n. prime n \<and> f n" shows "prime (find_prime f) \<and> f (find_prime f)"
simps: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" shows "wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> NS \<and> \<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> S then (True, True) else if (s, t) \<in> NS then if \<exists>i\<in>set (\<sigma> (f, length ss)). ss ! i \<succeq> t then (True, True) else case t of Var x \<Rightarrow> (False, ssimple \<and> large (f, length ss)) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (\<sigma> (g, length ts)). s \<succ> ts ! j) then if prs then (True, True) else let ss' = map ((!) ss) (\<sigma> (f, length ss)); ts' = map ((!) ts) (\<sigma> (g, length ts)); cf = c (f, length ss); cg = c (g, length ts) in if cf = Lex \<and> cg = Lex then lex_ext wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
inext_atLeastAtMost: fixes m :: "nat"   and t :: "nat"   and n :: "nat" assumes "m \<le> t"   and "t < n" shows "inext t {m..n} = Suc t"
BASE: fixes nsa :: "nat list"   and ia :: "nat"   and msa :: "nat list"   and ma :: "nat"   and ja :: "nat" assumes "\<And>n'. n' \<in> set nsa \<Longrightarrow> 0 < n'"   and "ia < b"   and "ia \<noteq> 0"   and "\<And>m'. m' \<in> set msa \<Longrightarrow> ma < m'"   and "ja < b"   and "ja \<noteq> 0"   and "ia * b ^ 0 + sum_list (map ((^) b) nsa) = ja * b ^ ma + sum_list (map ((^) b) msa)" shows "ia = ja \<and> 0 = ma"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and x0 :: "'c" assumes "t \<in> existence_ivl0 x0" shows "x0 \<in> Csafe"
Unit_def: fixes R :: "('d, 'e) Ring_scheme"   and a :: "'d" shows "Unit R a = (a \<in> carrier R \<and> (\<exists>b\<in>carrier R. a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = 1\<^sub>r\<^bsub>R\<^esub>))"
isTrue_all: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {x}"   and "\<forall>n\<in>num. isTrue (subst \<phi> n x)" shows "isTrue (all x \<phi>)"
kbo_supt_one: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and bef :: "('f, 'b) Term.term list"   and aft :: "('f, 'b) Term.term list" assumes "kbo'.NS s t" shows "kbo'.S (Fun f (bef @ s # aft)) t"
i_append_i_take_eq1: fixes n :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" assumes "n \<le> length xs" shows "(xs \<frown> f) \<Down> n = xs \<down> n"
i_set0_mult_empty: fixes k :: "nat" shows "{} \<in> i_set0_mult k"
sub_noteq_inf: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h"   and "w \<noteq> \<infinity>\<^sub>h" shows "z -\<^sub>h w \<noteq> \<infinity>\<^sub>h"
spd: shows "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"
induct: fixes P :: "Arity \<Rightarrow> exp \<Rightarrow> bool"   and a0 :: "Arity"   and a1 :: "exp" assumes "\<And>a e x. P (inc\<cdot>a) e \<Longrightarrow> P a (App e x)"   and "\<And>a x e. P (pred\<cdot>a) e \<Longrightarrow> P a (Lam [x]. e)"   and "\<And>a x. P a (Var x)"   and "\<And>a \<Gamma> e. \<lbrakk>\<And>x ea aa. (x, ea) \<in> set \<Gamma> \<Longrightarrow> P aa ea; P (fst (a, Aheap \<Gamma> e\<cdot>a)) e\<rbrakk> \<Longrightarrow> P a (Terms.Let \<Gamma> e)"   and "\<And>a b. P a (Bool b)"   and "\<And>a scrut e1 e2. \<lbrakk>P 0 scrut; P a e1; P a e2\<rbrakk> \<Longrightarrow> P a (scrut ? e1 : e2)" shows "P a0 a1"
cmod_eqI: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 * cnj z1 = z2 * cnj z2" shows "cmod z1 = cmod z2"
ex_prefix_of: fixes p :: "'b prefix" shows "\<exists>s. prefix_of p s"
word_ao_equiv: fixes w :: "'a word"   and w' :: "'a word" shows "(w = w OR w') = (w' = w AND w')"