hcju/isabelleps huggingface dataset:
the_resI: fixes S :: "'a dres"   and S' :: "'a nres" assumes "nres_of S \<le> S'"   and "S \<noteq> dSUCCEED" shows "RETURN (the_res S) \<le> S'"
local_flow_ball: fixes g :: "real" shows "local_flow (fball g) UNIV UNIV (\<lambda>t s. \<chi>i. if i = 1 then g \<cdot> t\<^sup>2 / 2 + s $ 2 \<cdot> t + s $ 1 else g \<cdot> t + s $ 2)"
finite_lists_distinct_length_eq: fixes A :: "'c set"   and n :: "nat" assumes "finite A" shows "finite {xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A}"
HEndPhase0_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2a s"   and "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2c_inner s p" shows "Inv2a s'"
fInf_distr: fixes f :: "'e \<Rightarrow> 'c" shows "Inf_pres (\<lambda>x. x \<circ> f)"
IH: shows "subst_floatarith (\<lambda>x. Var (s x)) (FDERIV_n_floatariths fas xs ds n ! i) = FDERIV_n_floatariths (map (subst_floatarith (\<lambda>x. Var (s x))) fas) xs (map (subst_floatarith (\<lambda>x. Var (s x))) ds) n ! i"
frontier_of_topspace: fixes X :: "'c topology" shows "X frontier_of topspace X = {}"
gen_contr_triv: fixes V :: "'a set" assumes "G.is_slim V"   and "pverts G = V" shows "gen_contr_graph (with_proj G) V = G"
1: fixes P :: "'a mat"   and A :: "'a mat"   and Q :: "'a mat" assumes "Domainp HMA_M3 (P, A, Q)" shows "P \<in> carrier_mat CARD('nr) CARD('nr) \<and> A \<in> carrier_mat CARD('nr) CARD('nc) \<and> Q \<in> carrier_mat CARD('nc) CARD('nc)"
oldest_token_always_def: fixes x :: "nat"   and n :: "nat" shows "\<exists>i\<le>x. oldest_token (token_run x n) n = Some i"
finsum_restrict: fixes f :: "'e \<Rightarrow> 'c"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "f \<in> A \<rightarrow> carrier V"   and "restrict f A = restrict g A" shows "finsum V f A = finsum V g A"
smult_l_null: fixes R :: "('c, 'd) ring_scheme"   and M :: "('c, 'e, 'f) module_scheme"   and x :: "'e" assumes "module R M"   and "x \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
WT_oracle2: fixes s2 :: "'s2" shows "\<I> \<turnstile>c oracle2 s2 \<surd>"
component_def: fixes F :: "'d program"   and H :: "'d program" shows "(F \<le> H) = (\<exists>G. F \<squnion> G = H)"
monad_resumption_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and pause :: "'f \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> 'e" shows "monad_resumption return bind pause \<equiv> monad return bind \<and> monad_resumption_axioms bind pause"
g_secure_def: fixes P :: "'a process"   and L :: "'a \<Rightarrow> g_level" shows "g_secure P L \<equiv> \<forall>xs x. xs @ [x] \<in> traces P \<and> L x = High \<longrightarrow> {ys'. \<exists>ys. xs @ ys \<in> traces P \<and> ys' = filter (\<lambda>y. L y = Low) ys} = {ys'. \<exists>ys. xs @ x # ys \<in> traces P \<and> ys' = filter (\<lambda>y. L y = Low) ys}"
fin_make_untimed_append: fixes x :: "'d list list"   and y :: "'d list list" assumes "fin_make_untimed x \<noteq> []" shows "fin_make_untimed (x @ y) \<noteq> []"
distinct_23: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PApp x31 x32 \<noteq> PFn x41 x42 x43"
solves_odeI: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set" assumes "(y has_vderiv_on (\<lambda>t. f t (y t))) T"   and "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X" shows "(y solves_ode f) T X"
lang_pow_subset_lists: fixes A :: "'b list set"   and S :: "'b set"   and n :: "nat" assumes "A \<subseteq> lists S" shows "A ^^ n \<subseteq> lists S"
Src_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "Src (Dom t) = Src t"
1: shows "one_final_but g (if u = v then {} else Edges (u # between (vertices f) u v @ [v]))"
LIM_zero_iff: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'b)) F = (f \<longlongrightarrow> l) F"
Q2_leq_lemma: shows "tensor_P (1\<^sub>m N) (mat_incr K) * Q2 * adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) \<le>\<^sub>L P' + Q"
add_diff_eq_ennreal: fixes z :: "ennreal"   and y :: "ennreal"   and x :: "ennreal" assumes "z \<le> y" shows "x + (y - z) = x + y - z"
elims_3: fixes x :: "'a extN"   and xa :: "'a extN" assumes "\<not> x < xa" obtains x :: "'a extN" and y :: "'a extN" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
S1_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S1 rmhist p = S rmhist False False False clkA rpcA histA histA p"
steps_alt_induct: fixes x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "G'.steps x"   and "\<And>x. P [x]"   and "\<And>y x xs. \<lbrakk>y \<rightarrow> x; G'.steps (xs @ [y]); P (xs @ [y])\<rbrakk> \<Longrightarrow> P (xs @ [y, x])" shows "P x"
comp_substs_empty_iff: fixes \<sigma>s :: "'s list"   and \<eta>s :: "'s list" shows "(\<sigma>s \<odot>s \<eta>s = []) = (\<sigma>s = [] \<or> \<eta>s = [])"
linorder_of_list_sorted_wrt_list_of_set: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" assumes "linorder_on A R"   and "finite A" shows "linorder_of_list (sorted_wrt_list_of_set R A) = R"
simps_1: fixes a :: "'b" shows "isJust\<cdot>(Just\<cdot>a) = TT"
compl_compl_eq_id: fixes s :: "'a set" shows "UNIV - (UNIV - s) = s"
rk2_fas'_def: fixes D :: "nat" shows "rk2_fas' D = map fold_const_fa (rk2_fas (Var (2 * D)) (map Var [0..<D]) (Var (2 * D + 1)) (map Var [D..<2 * D]) (Var (2 * D + 2)))"
assms_5: fixes Cs :: "'b set list" assumes "SimplicialComplex.min_maxsimpchain W (C # Cs)" shows "pgallery (f \<Turnstile> (C # Cs))"
proj_fst_k_def: fixes k :: "nat" shows "proj_fst_k k = mat K K (\<lambda>(i, j). if i = j \<and> i < k then 1 else 0)"
sum_diff_nat_ivl: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m \<le> n"   and "n \<le> p" shows "sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
assoc\<^sub>S_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set"   and c :: "'b local.term set" assumes "free_strict_monoidal_category C" shows "free_strict_monoidal_category.assoc\<^sub>S C a b c \<equiv> free_strict_monoidal_category.tensor\<^sub>S C a (free_strict_monoidal_category.tensor\<^sub>S C b c)"
rel_of_enex_def: fixes enex :: "('c \<Rightarrow> 'e set) \<times> ('e \<Rightarrow> 'c \<Rightarrow> 'd)" shows "rel_of_enex enex \<equiv> let (en, ex) = enex in {(s, ex a s) |s a. a \<in> en s}"
llength_ltake: fixes n :: "enat"   and xs :: "'c llist" shows "llength (ltake n xs) = min n (llength xs)"
list_combE_1: fixes f :: "'a"   and xs :: "'a list"   and S :: "name fset" assumes "closed_except (list_comb f xs) S" shows "closed_except f S"
classT: shows "class P Thread = \<lfloor>(T', fsT, MsT)\<rfloor>"
export_s: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and Q :: "'b assn" assumes "\<And>s'. G|-{\<lambda>Z s. s' = s \<and> P Z s}. c .{Q}" shows "G|-{P}. c .{Q}"
antecedent: shows "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))"
pdevs_apply_filter_pdevs: fixes I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a pdevs"   and i :: "nat" shows "pdevs_apply (filter_pdevs I x) i = (if I i (pdevs_apply x i) then pdevs_apply x i else (0::'a))"
supp_def: fixes x :: "'b" shows "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
npMulDistr: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat"   and m :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x^\<^bsup>R m\<^esup> = x^\<^bsup>R (n + m)\<^esup>"
measure_return: fixes X :: "'a set"   and M :: "'a measure"   and x :: "'a" assumes "X \<in> sets M" shows "Sigma_Algebra.measure (return M x) X = indicat_real X x"
has_field_def: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C has F,b:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>(b, T)\<rfloor>"
propex\<^sub>0: fixes F :: "\<o>" shows "\<exists>r. Some r = d\<^sub>0 F"
GPicard5: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on ball 0 1 - {0}"   and "\<And>z. z \<in> ball 0 1 - {0} \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1" obtains e :: "real" and B :: "real" where "0 < e" and "e < 1" and "0 < B" and "(\<forall>z\<in>ball 0 e - {0}. cmod (f z) \<le> B) \<or> (\<forall>z\<in>ball 0 e - {0}. B \<le> cmod (f z))"
NextSendUpdD_2: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and tt :: "('loc \<times> 't) set" assumes "NextSendUpd' c0 c1 p tt" shows "prop_config c1 = prop_config c0"
pure_tree_simps_3: fixes x :: "'b" shows "right (pure x) = pure x"
Red_F_imp_ex_non_Red_F: fixes C :: "'f"   and N :: "'f set" assumes "C \<in> Red_F N" shows "\<exists>CC\<subseteq>N - Red_F N. CC \<Turnstile> {C} \<and> (\<forall>C'\<in>CC. C' < C)"
refl: fixes a :: "'a::order" shows "(a::'a::order) \<le> a"
not_tendsto_and_filterlim_at_infinity: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> c) F"   and "filterlim f at_infinity F" shows "False"
paths_in_all_paths: shows "paths \<subseteq> all_paths"
r_into_trancl': fixes p :: "'a \<times> 'a"   and r :: "('a \<times> 'a) set" assumes "p \<in> r" shows "p \<in> r\<^sup>+"
diff_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = inf x (- y)"
formulas_def: fixes S :: "'b set" shows "formulas S = {x. \<forall>A B. (\<forall>i\<in>S. (i \<in> A) = (i \<in> B)) \<longrightarrow> (A \<in> Rep_formula x) = (B \<in> Rep_formula x)}"
Sup_set_def: fixes A :: "'e set set" shows "\<Union> A = {x. Sup ((\<in>) x ` A)}"
lesssub_def: fixes x :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" shows "(x \<sqsubset>\<^bsub>r\<^esub> y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> x \<noteq> y)"
finite_input: fixes nid :: "'nid"   and t :: "'a \<Rightarrow> cnf"   and n :: "'a" assumes "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>" shows "finite (pin (\<sigma>\<^bsub>nid\<^esub>t n))"
swapEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and x :: "'var" assumes "goodEnv rho" shows "freshEnv xs (x @xs[y1 \<and> y2]_ys) (rho &[y1 \<and> y2]_ys) = freshEnv xs x rho"
index_oddpart_unique: fixes m :: "nat"   and m' :: "nat"   and i :: "nat"   and i' :: "nat" assumes "odd m"   and "odd m'" shows "(2 ^ i * m = 2 ^ i' * m') = (i = i' \<and> m = m')"
set_inorder: fixes t :: "'b tree" shows "set (inorder t) = set_tree t"
fold_insert_fset: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'c"   and A :: "'c fset" shows "fold_fset f (insert_fset a A) = (if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
az: shows "rtrancl_path r a (as @ y # ys) z"
SetMark_correct1: shows "Hoare_dgr SetMarkInv SetMark (SetMarkInv \<sqinter> - grd (step SetMark))"
gvcon: shows "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
finite_edge_set: shows "finite E\<^bsub>H\<^esub>"
assms: shows "k < length (trace_parallel_plan_sas_plus I \<psi>) - 1"
LIMSEQ_offset: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a" assumes "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" shows "f \<longlonglongrightarrow> a"
map: shows "\<beta> \<in> Partition \<rightarrow>\<^sub>E T"
worklist_aux_Cons: fixes succs :: "'d \<Rightarrow> 'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c"   and ws' :: "'c list"   and s :: "'d" shows "worklist_aux succs f (x # ws', s) = worklist_aux succs f (succs s x @ ws', f x s)"
sub_non_negative: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (neg_numeral.sub minus plus one n m) = (m \<le> n)"
upper_bound_E: shows "\<exists>E. \<forall>n. B (2 * n) \<le> E * B n"
some_counit_def: fixes f :: "'a" shows "some_counit f \<equiv> SOME \<epsilon>. adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* (some_unit f) \<epsilon>"
word_ao_dist: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "(x OR y) AND z = x AND z OR y AND z"
I: shows "f (Suc (nat (padic_val p f + padic_val p g))) = i * p ^ nat (padic_val p f) \<and> \<not> p dvd int (nat i)"
less_eq_eint_def: fixes m :: "eint"   and n :: "eint" shows "(m \<le> n) = (case n of eint n1 \<Rightarrow> case m of eint m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False | \<infinity> \<Rightarrow> True)"
assoc_coherence: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "G \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi>\<^sub>G (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi>\<^sub>G (f, g) \<star>\<^sub>D G h) = \<Phi>\<^sub>G (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Phi>\<^sub>G (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[G f, G g, G h]"
preserves_ide: fixes a :: "'a" assumes "C.ide a" shows "C.ide (trg\<^sub>C a)"
assms: shows "X \<subseteq>\<^sub>\<circ> Y"
funcset_image: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
bounded_linear_ident: shows "bounded_linear (\<lambda>x. x)"
rs_3: shows "(label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i) \<noteq> (label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i)"
lc_monic: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "lc (monic p) = (1::'c)"
distrib_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y \<cdot> z = (x + y) \<cdot> (x + z)"
in_homE: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" obtains "J.arr f" and "J.dom f = a" and "J.cod f = b"
inter_plus_addition_notin: fixes m\<^sub>1 :: "('d, 'c) fmap"   and k :: "'d"   and m\<^sub>2 :: "('d, 'c) fmap"   and v :: "'c" assumes "m\<^sub>1 $$ k = None"   and "m\<^sub>2 $$ k = None" shows "m\<^sub>1(k $$:= v) \<inter>\<^sub>+ m\<^sub>2 = m\<^sub>1 \<inter>\<^sub>+ m\<^sub>2"
inj_monoid_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_monoid_add_hom hom \<equiv> monoid_add_hom hom \<and> inj_semigroup_add_hom hom"
h'_def: shows "h' = (\<lambda>s t. exp (complex_of_real (a * r) * cis t - (complex_of_real (ln r) + \<i> * complex_of_real t) * s) * (complex_of_real (- ln r) - \<i> * complex_of_real t) * (complex_of_real r * \<i> * cis t) / (1 - exp (complex_of_real r * cis t)))"
select_convs_1: fixes verts :: "'a set"   and arcs :: "('a \<times> 'a) set"   and more :: "'b" shows "V\<^bsub>\<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = verts"
yf: shows "YFtor C @@ X = Hom\<^bsub>C\<^esub>[\<midarrow>,X]"
y_split: shows "as1 = bs @ [(yid, yr)] @ cs \<and> (\<forall>i\<in>set (map fst bs). i < yid) \<and> (\<forall>i\<in>set (map fst cs). yid < i)"
jumpF_polyR_mult_cancel: fixes p' :: "real poly"   and q :: "real poly"   and p :: "real poly"   and a :: "real" assumes "p' \<noteq> 0" shows "jumpF_polyR (p' * q) (p' * p) a = jumpF_polyR q p a"
of_int_lessD: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "int"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (abs (ring_1.of_int one plus zero minus n)) x" shows "n = 0 \<or> less one x"
seqI: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.arr f"   and "S.arr g"   and "S.dom g = S.cod f" shows "S.seq g f"
elims_2: fixes x :: "(Vars.var \<times> 'b) list \<times> stack_elem list" assumes "heap_upds_ok x" obtains \<Gamma> :: "(Vars.var \<times> 'b) list" and S :: "stack_elem list" where "x = (\<Gamma>, S)" and "domA \<Gamma> \<inter> upds S = {} \<and> distinct (upds_list S)"
power_mono_odd: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "odd n"   and "less_eq a b" shows "less_eq (power.power one times a n) (power.power one times b n)"
ide_dom: fixes f :: "'a" assumes "C.arr f" shows "C.ide (C.dom f)"
Carmichael_prime: fixes p :: "nat" assumes "prime p" shows "Carmichael p = p - 1"
subst_restrict: fixes t :: "'a"   and M :: "name fset"   and env :: "(name, 'a) fmap" assumes "frees t |\<subseteq>| M" shows "subst t (fmrestrict_fset M env) = subst t env"
set_map_fst: shows "set (map fst \<Gamma>) = dom (map_of \<Gamma>)"
AllowedActs_totalize: fixes F :: "'c program" shows "AllowedActs (totalize F) = AllowedActs F"
p_res_ring_one: fixes k :: "nat" assumes "0 < k" shows "\<one>\<^bsub>residue_ring (p ^ k)\<^esub> = 1"
FunctorCompDef: fixes f :: "'c"   and g :: "'c" assumes "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f \<approx>>\<^bsub>CatCod F\<^esub> F ## g"
fun_rel_comp_dist: fixes R1 :: "('c \<times> 'g) set"   and R2 :: "('d \<times> 'h) set"   and R3 :: "('g \<times> 'e) set"   and R4 :: "('h \<times> 'f) set" shows "(R1 \<rightarrow> R2) O (R3 \<rightarrow> R4) \<subseteq> R1 O R3 \<rightarrow> R2 O R4"
cut_greater_card: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "card (I \<down>> t) \<le> card I"
assms_6: shows "a \<oplus>\<^bsub>padic_int p\<^esub> b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
int_to_bv_ge0: fixes n :: "int" assumes "0 \<le> n" shows "int_to_bv n = norm_signed (\<zero> # nat_to_bv (nat n))"
simps_1: fixes r :: "real" shows "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
char_Pg_plus: fixes g :: "game"   and h :: "game" shows "Abs_Pg (eq_game_rel `` {g}) + Abs_Pg (eq_game_rel `` {h}) = Abs_Pg (eq_game_rel `` {plus_game g h})"
matrix_sum_add: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat"   and g :: "nat \<Rightarrow> 'a mat"   and h :: "nat \<Rightarrow> 'a mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> h k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k = g k + h k" shows "matrix_sum d f n = matrix_sum d g n + matrix_sum d h n"
subclsSp_subclsS_eq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "subclsSp P = subclsS' P"
a_get_dom_component_def: fixes get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and ptr :: "(_) object_ptr" shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component get_root_node to_tree_order ptr = Heap_Error_Monad.bind (get_root_node ptr) to_tree_order"
simplex_in_max: fixes X :: "'b set set"   and y :: "'b set" assumes "ChamberComplex X"   and "y \<in> X" shows "\<exists>x. SimplicialComplex.maxsimp X x \<and> order.greater_eq x y"
inv_eq_neg_one_eq: fixes x :: "int" assumes "x \<in> Units R" shows "(inv x = \<ominus> \<one>) = (x = \<ominus> \<one>)"
split_float_intervalD: fixes X :: "float interval"   and A :: "float interval"   and B :: "float interval" assumes "split_float_interval X = (A, B)" shows "set_of X \<subseteq> set_of A \<union> set_of B"
sub_2: shows "S \<subseteq> \<Union> \<D>"
OclAnd_idem: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and X) = X"
is_mgu_def: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'd) Term.term"   and E :: "(('e, 'd) Term.term \<times> ('e, 'd) Term.term) set" shows "is_mgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<exists>\<gamma>. \<tau> = \<sigma> \<circ>\<^sub>s \<gamma>))"
polychain_of_eq_empty_iff: fixes p :: "'a"   and xs :: "'a list" shows "(polychain_of p xs = []) = (xs = [])"
insert_eqpoll_insertD: fixes u :: "'a"   and A :: "'a set"   and v :: "'b"   and B :: "'b set" assumes "insert u A \<approx> insert v B"   and "u \<notin> A"   and "v \<notin> B" shows "A \<approx> B"
language_def: shows "ST.language \<equiv> {xs. ST.nextl (dfa.init \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>) xs \<in> dfa.final \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>}"
all_valid_fold: fixes G :: "'b gexp list"   and s :: "'b \<Rightarrow> value option" assumes "\<forall>g\<in>set G. gval g s \<noteq> invalid" shows "gval (fold gAnd G (Bc True)) s \<noteq> invalid"
assms_4: shows "\<pi>\<^sub>5 (the (rt dip)) = \<pi>\<^sub>5 (the (rt' dip))"
subgroup_axioms: shows "subgroup (translation ` H) transformation.Sym (compose G) (identity G)"
slice_def: fixes n :: "nat" shows "slice n = slice1 (LENGTH('b) - n)"
finite_vars_1: shows "finite {x \<in> vars\<^sub>s\<^sub>t S. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<and> x \<notin> set X}"
assms: shows "\<not> (reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0)"
trimmed_ms_aux_imp_nz: fixes basis :: "(real \<Rightarrow> real) list"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux xs f basis"   and "trimmed_ms_aux xs" shows "\<forall>\<^sub>F x in at_top. f x \<noteq> 0"
pop_aassoc: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a \<plusminus>\<^bsub>R\<^esub> b \<plusminus>\<^bsub>R\<^esub> c = a \<plusminus>\<^bsub>R\<^esub> (b \<plusminus>\<^bsub>R\<^esub> c)"
simps: fixes a1 :: "'h"   and a2 :: "'i"   and r :: "('h \<times> 'j) set"   and s :: "('j \<times> 'i) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
continuous_map_image_subset_topspace: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
is_empty_holds: fixes P :: "unit pred" shows "is_empty P = (\<not> holds P)"
rbl_word_plus: fixes x :: "'a word"   and y :: "'a word" shows "rev (to_bl (x + y)) = rbl_plus False (rev (to_bl x)) (rev (to_bl y))"