hcju/isabelleps huggingface dataset:
spoly_rep_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and g1 :: "'t \<Rightarrow>\<^sub>0 'c"   and g2 :: "'t \<Rightarrow>\<^sub>0 'c" shows "spoly_rep d m G g1 g2 = (\<exists>q. spoly g1 g2 = (\<Sum>g\<in>G. q g \<odot> g) \<and> (\<forall>g. q g \<in> punit.dgrad_p_set d m \<and> (q g \<odot> g \<noteq> 0 \<longrightarrow> lt (q g \<odot> g) \<prec>\<^sub>t term_of_pair (lcs (lp g1) (lp g2), component_of_term (lt g2)))))"
fmdrop_idle: fixes x :: "'d"   and B :: "('d, 'e) fmap" assumes "x |\<notin>| fmdom B" shows "fmdrop x B = B"
fct_gen_subalgebra_eq_sets: fixes M :: "'c measure"   and P :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd" assumes "sets M = sets P" shows "fct_gen_subalgebra M N X = fct_gen_subalgebra P N X"
set_update_subset_insert: fixes xs :: "'g list"   and i :: "nat"   and x :: "'g" shows "set (xs[i := x]) \<subseteq> insert x (set xs)"
in_hom_def: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "VV.in_hom f a b \<equiv> a \<in> VV.domains f \<and> b \<in> VV.codomains f"
simps_3: fixes t :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" shows "eval_expr t (Tmp sop) = sop"
inj_combinator: fixes x :: "'c"   and S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" assumes "x \<notin> S" shows "inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
Ia: fixes a :: "nat" assumes "0 < a" shows "u (a * N) x \<le> (\<Sum>i<a. u N ((T ^^ (i * N)) x))"
epred_eSuc: fixes n :: "enat" shows "epred (eSuc n) = n"
IH_4: fixes x :: "('a, 'b) rbt"   and y :: "('a \<times> 'b) list"   and x21 :: "'a \<times> 'b"   and x22 :: "('a \<times> 'b) list"   and xa :: "'a"   and ya :: "'b" assumes "\<not> (n = 0 \<or> n = 1)"   and "n mod 2 \<noteq> 0"   and "(x, y) = rbtreeify_f (n div 2) kvs"   and "y = x21 # x22"   and "(xa, ya) = x21"   and "0 < n div 2"   and "n div 2 \<le> Suc (length x22)" shows "length (snd (rbtreeify_g (n div 2) x22)) = Suc (length x22) - n div 2"
hlde_def: fixes a :: "nat list"   and b :: "nat list" shows "hlde a b \<equiv> 0 \<notin> set a \<and> 0 \<notin> set b"
a1: shows "low_equal s1 s2 \<and> t1 = s1\<lparr>traps := new_traps\<rparr> \<and> t2 = s2\<lparr>traps := new_traps\<rparr>"
in_arcs_add_arc_iff: fixes a :: "'b"   and u :: "'a" shows "in_arcs (add_arc a) u = (if head G a = u then insert a (in_arcs G u) else in_arcs G u)"
hintikka_def: fixes H :: "('a, 'b) form set" shows "hintikka H = ((\<forall>p ts. \<not> (Pred p ts \<in> H \<and> Neg (Pred p ts) \<in> H)) \<and> FF \<notin> H \<and> Neg TT \<notin> H \<and> (\<forall>Z. Neg (Neg Z) \<in> H \<longrightarrow> Z \<in> H) \<and> (\<forall>A B. And A B \<in> H \<longrightarrow> A \<in> H \<and> B \<in> H) \<and> (\<forall>A B. Neg (Or A B) \<in> H \<longrightarrow> Neg A \<in> H \<and> Neg B \<in> H) \<and> (\<forall>A B. Or A B \<in> H \<longrightarrow> A \<in> H \<or> B \<in> H) \<and> (\<forall>A B. Neg (And A B) \<in> H \<longrightarrow> Neg A \<in> H \<or> Neg B \<in> H) \<and> (\<forall>A B. Impl A B \<in> H \<longrightarrow> Neg A \<in> H \<or> B \<in> H) \<and> (\<forall>A B. Neg (Impl A B) \<in> H \<longrightarrow> A \<in> H \<and> Neg B \<in> H) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> H \<longrightarrow> P[t/0] \<in> H) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> H \<longrightarrow> Neg (P[t/0]) \<in> H) \<and> (\<forall>P. Exists P \<in> H \<longrightarrow> (\<exists>t. closedt 0 t \<and> P[t/0] \<in> H)) \<and> (\<forall>P. Neg (Forall P) \<in> H \<longrightarrow> (\<exists>t. closedt 0 t \<and> Neg (P[t/0]) \<in> H)))"
existence_ivl_maximal_segment: fixes x :: "real \<Rightarrow> 'a"   and t0 :: "real"   and t :: "real"   and x0 :: "'a" assumes "(x solves_ode g) {t0--t} Y"   and "x t0 = x0"   and "{t0--t} \<subseteq> S" shows "t \<in> Y.existence_ivl t0 x0"
trivial_relative_homology_group: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" assumes "p < 0" shows "trivial_group (relative_homology_group p X S)"
Amicable_pair_equiv_def_conv: fixes m :: "nat"   and n :: "nat" assumes "1 \<le> m"   and "1 \<le> n"   and "Esigma m = Esigma n \<and> Esigma m = m + n" shows "m Amic n"
pos_root_exI: fixes p :: "real poly" assumes "poly p 0 * lead_coeff p < 0" obtains x :: "real" where "0 < x" and "poly p x = 0"
BisT_MtransT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>T d"   and "mustT c s"   and "mustT d t"   and "s \<approx> t"   and "(c, s) \<rightarrow>*t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'"
row_space_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "row_space A = {w. \<exists>y. Finite_Cartesian_Product.transpose A *v y = w}"
get_ancestors_di_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_ancestors_di ptr" shows "ptr |\<in>| object_ptr_kinds h"
interleave_set: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "zs \<in> xs \<otimes> ys" shows "set zs = set xs \<union> set ys"
simps_2: fixes cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and v0 :: "v"   and p :: "pat"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "cupcake_match_result cenv v0 ((p, e) # pes) err_v = (if distinct (pat_bindings p []) then case cupcake_pmatch cenv p v0 [] of No_match \<Rightarrow> cupcake_match_result cenv v0 pes err_v | Match_type_error \<Rightarrow> Rerr (Rabort Rtype_error) | Match env' \<Rightarrow> Rval (e, p, env') else Rerr (Rabort Rtype_error))"
add_rel_class_of: fixes U :: "'c set"   and x :: "'d"   and V :: "'c set"   and y :: "'d"   and W :: "'c set" assumes "(U, x) \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y"   and "(V, y) \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y"   and "W \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stfx.add_rel (stfx.class_of U x) (stfx.class_of V y) = stfx.class_of W (add_str\<^sub>Y W (\<rho>\<^sub>Y U W x) (\<rho>\<^sub>Y V W y))"
RETURN_SPEC_refine: fixes x :: "'c"   and R :: "('c \<times> 'a) set"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'" shows "RETURN x \<le> \<Down> R (SPEC \<Phi>)"
intro: fixes G :: "('e, 'f) Graph_scheme" assumes "Graph G"   and "Tree_axioms G" shows "Tree G"
after_summary_nonneg: fixes M :: "'t zmultiset"   and S :: "'sum antichain"   and t :: "'t" assumes "\<And>t. 0 \<le> zcount M t" shows "0 \<le> zcount (after_summary M S) t"
matrix_to_iarray_interchange_rows: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "matrix_to_iarray (interchange_rows A i j) = interchange_rows_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
case_b: shows "\<turnstile> (P \<inter> b) c1 Q"
is_natural_2: fixes f :: "'b" assumes "arr f" shows "\<ll>'.map (cod f) \<cdot> local.map f = \<ll>'.map f"
apex_def: shows "\<nu>.dom.apex \<equiv> C.dom \<nu>.dom.leg0"
wlp_def: fixes pr :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wlp pr \<equiv> pr False"
chart_in_codomain: fixes x :: "'b"   and c :: "('b, 'c) chart" assumes "x \<in> domain c" shows "apply_chart c x \<in> codomain c"
bex_empty: fixes P :: "'b::type \<Rightarrow> bool" shows "Bex {} (P::'b::type \<Rightarrow> bool) = False"
distinct_mset_minus: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "distinct_mset A" shows "distinct_mset (A - B)"
sgn_QuantT_def: fixes x :: "'a['b, 'c]" shows "sgn x = x /\<^sub>R norm x"
\<delta>_states_accsI: fixes \<delta> :: "('a, 'b) ta_rule set"   and n :: "'b tree"   and q :: "'a" assumes "accs \<delta> n q" shows "q \<in> \<delta>_states \<delta>"
plossless_converter_mono: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and \<I>1' :: "('i, 'j) \<I>"   and \<I>2' :: "('k, 'l) \<I>" assumes "plossless_converter \<I>1 \<I>2 conv"   and "outs_\<I> \<I>1' \<subseteq> outs_\<I> \<I>1"   and "\<I>2 \<le> \<I>2'"   and "\<I>1, \<I>2 \<turnstile>\<^sub>C conv \<surd>" shows "plossless_converter \<I>1' \<I>2' conv"
app_seqsimilar: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "s \<approx> t" shows "x ## s \<approx> x ## t"
simps: fixes bs :: "bool list"   and "is" :: "int list"   and i :: "int"   and d :: "int" shows "lderiv0 bs (presb.Eq is i d) = (if d \<noteq> 0 then undefined else let v = i - scalar_product (map of_bool bs) is in if v mod 2 = 0 then FBase (presb.Eq is (v div 2) 0) else FBool False)"
nf: fixes S :: "'N set" assumes "convex S"   and "(0::'N) \<in> S" shows "negligible (frontier S)"
linorder_axioms_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.linorder_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<or> less_eq y x"
Wbis_converse: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "((c, d) \<in> theta\<inverse> \<union> Wbis) = ((d, c) \<in> theta \<union> Wbis)"
preserves_inv: fixes f :: "'a" assumes "local.iso f" shows "R (local.inv f) = local.inv (R f)"
hom_deg_set_def: fixes z :: "nat"   and A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "hom_deg_set z A = (\<lambda>a. hom_component a z) ` A"
subst_cong: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<sigma>' :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('b, 'd) Term.term"   and \<theta>' :: "'c \<Rightarrow> ('b, 'd) Term.term" assumes "\<sigma> = \<sigma>'"   and "\<theta> = \<theta>'" shows "\<sigma> \<circ>\<^sub>s \<theta> = \<sigma>' \<circ>\<^sub>s \<theta>'"
Nml_HcompNml_4: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Src (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Src u"
hyps_3: shows "sh' C = \<lfloor>(sfs, i)\<rfloor>"
generatorE: fixes G :: "('c, 'd) cyclic_group_scheme"   and x :: "'c" assumes "cyclic_group G"   and "x \<in> carrier G" obtains n :: "nat" where "x = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> n"
31: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<squnion> y) = x \<squnion> y"
assms_2: assumes "\<exists>p l. P l p" shows "Q"
fps_sub_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f - g) n = fps_nth f n - fps_nth g n"
alphaAbs_qAbs_iff_all_distinct_qAFresh: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
lunit_in_hom_1: fixes f :: "'a" assumes "ide f" shows "arr (PB.lunit f) \<and> src f \<in> sources (PB.lunit f) \<and> trg f \<in> targets (PB.lunit f)"
config_n_init: fixes s0 :: "'c list"   and qs :: "'c list" shows "map_pmf (snd \<circ> snd) (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs) = return_pmf s0"
sin_Taylor_0_upper_bound_pos: fixes x :: "real" assumes "0 \<le> x" shows "sin x \<le> x"
Keys_range_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "Keys (Poly_Mapping.range (vectorize_poly p)) = pp_of_term ` keys p"
ennreal_mult_top: fixes a :: "ennreal" shows "a * top = (if a = 0 then 0 else top)"
finite_syms: fixes s :: "('c, 'd) tm" shows "finite (syms s)"
elims: fixes x :: "'b"   and xa :: "nat"   and y :: "nat option" assumes "state_rank x xa = y" obtains q :: "'b" and n :: "nat" where "x = q" and "xa = n" and "y = (if configuration q n \<noteq> {} \<and> \<not> sink q then Some (card (senior_states q n)) else None)"
Nil_in_conc: fixes A :: "'b list set"   and B :: "'b list set" shows "([] \<in> A \<cdot> B) = ([] \<in> A \<and> [] \<in> B)"
cnj_mult: fixes x :: "octo"   and y :: "octo" shows "Octonions.cnj (x * y) = Octonions.cnj y * Octonions.cnj x"
rel_def: fixes Rv :: "('V \<times> 'b) set"   and Rw :: "('W \<times> 'c) set" shows "\<langle>Rv, Rw\<rangle>rel \<equiv> br \<alpha> invar O \<langle>Rv, Rw\<rangle>graph_rel"
foldr_filter: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'h \<Rightarrow> bool"   and xs :: "'h list" shows "foldr f (filter P xs) = foldr (\<lambda>x. if P x then f x else id) xs"
map\<^sub>0_simps_2: fixes a :: "'d" assumes "D.obj a" shows "src\<^sub>D (GF.map\<^sub>0 a) = GF.map\<^sub>0 a"
strict_mono_to_nat: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "order.strict_mono less (mod_type.to_nat Rep)"
elims: fixes x :: "'c DBMEntry"   and xa :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "dbm_add x xa = y" obtains uu_ :: "'c DBMEntry" where "x = \<infinity>" and "xa = uu_" and "y = \<infinity>"   | v :: "'c" where "x = Le va__" and "xa = \<infinity>" and "y = \<infinity>"   | v :: "'c" where "x = Lt v" and "xa = \<infinity>" and "y = \<infinity>"   | a :: "'c" and b :: "'c" where "x = Le a__" and "xa = Le b__" and "y = Le (a__ + b__)"   | a :: "'c" and b :: "'c" where "x = Le aa__" and "xa = Lt ba__" and "y = Lt (aa__ + ba__)"   | a :: "'c" and b :: "'c" where "x = Lt ab__" and "xa = Le bb__" and "y = Lt (ab__ + bb__)"   | a :: "'c" and b :: "'c" where "x = Lt a" and "xa = Lt b" and "y = Lt (a + b)"
var_rename_wt'_1: fixes \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and m :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and n :: "nat" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "s = m \<cdot> \<delta>" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (var_rename_inv n \<circ>\<^sub>s \<delta>)"
elims_3: fixes x :: "nat"   and xa :: "('a \<times> bool \<times> 'b tree23) option" assumes "\<not> dfull x xa" obtains n :: "nat" and p :: "'a" and t' :: "'b tree23" where "x = n__" and "xa = Some (p__, True, t'__)" and "\<not> full n__ t'__"   | n :: "nat" and p :: "'a" and t' :: "'b tree23" where "x = n" and "xa = Some (p, False, t')" and "\<not> full (Suc n) t'"
in_keys_timesE: fixes v :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (p * q)" obtains t :: "'b" and u :: "'b" where "t \<in> keys p" and "u \<in> keys q" and "v = t + u"
clo0T: shows "closedin (top_of_set ({0..1} \<times> T)) ({0} \<times> T)"
is_singleton_the_elem: fixes A :: "'d set" shows "is_singleton A = (A = {the_elem A})"
semiring_1_ow_def: fixes U :: "'a set"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" shows "semiring_1_ow U one times plus zero \<equiv> zero_neq_one_ow U one zero \<and> semiring_0_ow U plus zero times \<and> monoid_mult_ow U one times"
x0_hb_bound6: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "x * ((1 - bs ! i) / 2) \<le> x - (bs ! i * x + (hs ! i) x)"
dependent_single: fixes x :: "'d" shows "cdependent {x} = (x = (0::'d))"
essential_weight_update: fixes f :: "('k \<Rightarrow> ennreal) \<Rightarrow> 'k \<Rightarrow> ennreal"   and G :: "('k, 'l) web_scheme" shows "essential (weight_update f G) = essential G"
\<T>covered: fixes K :: "'a set" assumes "K \<in> \<T>" shows "\<exists>C. C \<in> \<N> \<and> K \<subseteq> C"
atom_in_wm_eq: fixes s :: "predicate \<times> object list \<Rightarrow> bool"   and atm :: "object atom" shows "close_eq s \<Turnstile> Atom atm = (Atom atm \<in> close_eq_M (state_to_wm s))"
simps_1: fixes uu :: "real_max_bot" shows "(uu < MInfty) = False"
domsMT_notND_DD3: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "x \<in> dom (C (DenyAllFromTo a b))"   and "x \<in> dom (C (DenyAllFromTo c d))" shows "\<not> netsDistinct a c"
graph_axioms: fixes m :: "'a" shows "graph (to_gbg_ext m)"
LowDom_def: fixes F :: "(Field \<times> Val) list" shows "LowDom F = {f. \<exists>v. lookup F f = Some v \<and> GAMMA f = low}"
stabiliser_cosets_equivalent: fixes g :: "Vertex \<Rightarrow> Vertex"   and h :: "Vertex \<Rightarrow> Vertex"   and x :: "Vertex" assumes "g \<in> carrier tetrahedral_group"   and "h \<in> carrier tetrahedral_group" shows "(apply_rotation g x = apply_rotation h x) = (g <#\<^bsub>tetrahedral_group\<^esub> tetrahedral.stabiliser x = h <#\<^bsub>tetrahedral_group\<^esub> tetrahedral.stabiliser x)"
h_plus_not_infty: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_plus i e \<noteq> \<top>"
a_get_component_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_component ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
cgenideal_def: fixes R :: "('a, 'b) monoid_scheme"   and a :: "'a" shows "PIdl\<^bsub>R\<^esub> a = {x \<otimes>\<^bsub>R\<^esub> a |x. x \<in> carrier R}"
multI: fixes r :: "('b \<times> 'b) set"   and M :: "'b multiset"   and I :: "'b multiset"   and K :: "'b multiset"   and N :: "'b multiset"   and J :: "'b multiset" assumes "trans r"   and "M = I + K"   and "N = I + J"   and "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(M, N) \<in> mult r"
intro: fixes grouping :: "int"   and point_set :: "'b set"   and block_collection :: "'b set multiset"   and index :: "int" assumes "1 \<le> grouping"   and "grouping \<le> int (card point_set)"   and "\<And>ps. \<lbrakk>ps \<subseteq> point_set; int (card ps) = grouping\<rbrakk> \<Longrightarrow> int (block_collection index ps) = index" shows "t_wise_balance_axioms point_set block_collection grouping index"
homeomorphic_empty_1: fixes S :: "'a set" shows "S homeomorphic {} = (S = {})"
assms_1: fixes x :: "'a" assumes "x \<in> space M" shows "sets (f x) = sets N"
2: shows "\<turnstile> DBSpec \<longrightarrow> \<box>[FullNxt]_vars"
neq0_conv: fixes n :: "V" shows "(n \<noteq> 0) = (0 < n)"
basis_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>'\<^sub>1.basis_in \<E> X = indep_system.basis \<E> (\<E>'\<^sub>1.indep_in \<E>) X"
b: fixes k :: "nat" shows "LINT x|M. f ((T ^^ k) x) = integral\<^sup>L M f"
2: fixes is1 :: "nat list"   and is2 :: "nat list" assumes "is1 \<lhd> input_sizes m1"   and "is2 \<lhd> input_sizes m2" shows "lookup (tensors_from_net (Pool m1 m2) $ j) (is1 @ is2) = lookup (tensors_from_net m1 $ j) is1 * lookup (tensors_from_net m2 $ j) is2"
unit_simps_3: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "local.dom \<i>[a] = a \<star> a"
td_ext_axioms_def: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and norm :: "'b \<Rightarrow> 'b" shows "td_ext_axioms Rep Abs norm \<equiv> \<forall>x. Rep (Abs x) = norm x"
order_trans: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
in_homE: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "S.arr f" and "S.dom f = a" and "S.cod f = b"
is_rbt_impl_of: fixes t :: "('a, 'b) RBT.rbt" shows "is_rbt (impl_of t)"
DERIV_def: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b" shows "(f has_field_derivative D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'b\<rightarrow> D"
strict_iff_order: fixes b :: "'k multiset"   and a :: "'k multiset" shows "(b \<subset># a) = (b \<subseteq># a \<and> a \<noteq> b)"
divmod_algorithm_code_7: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "divmod (num.Bit0 m) (num.Bit1 n) = (if m \<le> n then (zero, numeral.numeral one plus (num.Bit0 m)) else divmod_step (num.Bit1 n) (divmod (num.Bit0 m) (num.Bit0 (num.Bit1 n))))"
beukers_denom_ineq: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x \<in> {0<..<1}"   and "y \<in> {0<..<1}"   and "z \<in> {0..1}" shows "(1 - x * y) * z < 1"
distinct_3: fixes x11 :: "exp"   and x12 :: "exp"   and x3 :: "Vars.var" shows "Alts x11 x12 \<noteq> Upd x3"
insert: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
assms_1: shows "(f has_real_derivative f') (at x within A)"
orbit_eq_singleton_iff: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(orbit f x = {x}) = (f x = x)"
netmap_is_fst_netgmap': fixes s' :: "'c net_state"   and i :: "nat"   and s :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e" assumes "netmap s' i = netmap s i" shows "fst (netgmap sr s') i = fst (netgmap sr s) i"
a_Unauth: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : AuthT \<tau>" shows "\<Gamma> \<turnstile> Unauth e, Unauth eP, Unauth eV : \<tau>"
eq_Max_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(m = linorder.Max (\<sqsubseteq>) A) = (m \<in> A \<and> (\<forall>a\<in>A. a \<sqsubseteq> m))"
combine_eq_eval: fixes Ks :: "'a list list"   and x :: "'a list" shows "UP.combine Ks (UP.exp_base x (length Ks)) = UP.eval Ks x"
zero_r_neutral_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b" shows "zero_r_neutral f \<equiv> \<forall>a. f a (0::'c) = a"
r: shows "bounded ((\<lambda>z. f z - a) ` (ball z r - {z})) \<or> bounded ((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
prems_1: shows "ccw'.sortedP 0 (z # zs)"
nc_iff2: fixes x :: "('a \<times> 'a set) set" shows "(x \<subseteq> NC) = (x \<cdot> {} = {})"
FIX_def: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "FIX \<phi> = (\<lambda>(s, t, \<beta>). \<forall>\<Phi>. (\<forall>ss tt \<gamma>. \<phi> \<Phi> (ss, tt, \<gamma>) \<longrightarrow> \<Phi> (ss, tt, \<gamma>)) \<longrightarrow> \<Phi> (s, t, \<beta>))"
to_pre_digraph_def: fixes G :: "('c, 'd) Graph_scheme" shows "to_pre_digraph G \<equiv> \<lparr>pre_digraph.verts = V\<^bsub>G\<^esub>, arcs = E\<^bsub>G\<^esub>, tail = fst, head = snd\<rparr>"