hcju/isabelleps huggingface dataset:
MonotProblem_axioms: shows "MonotProblem TE_wtFsym wtPsym TE_arOf TE_resOf parOf tPB"
changes_poly_at_0_1: fixes a :: "'a" shows "changes_poly_at [] a = 0"
simps_3: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and m :: "'b common_primitive match_expr"   and rs :: "'b common_primitive rule list"   and allowed :: "'b word set"   and denied1 :: "'b word set" shows "no_spoofing_algorithm iface ipassmt (Rule m Drop # rs) allowed denied1 = no_spoofing_algorithm iface ipassmt rs allowed (denied1 \<union> (get_all_matching_src_ips iface m - allowed))"
tendsF: fixes x :: "'a" assumes "x \<notin> N__" shows "(\<lambda>n. F n x) \<longlonglongrightarrow> f x"
NOTLEFT_eqvt_name: fixes pi :: "(name \<times> name) list"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> NOTLEFT (NOT B) X = NOTLEFT (NOT B) (pi \<bullet> X)"
is_num_normalize_1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "plus (plus a b) c = plus a (plus b c)"
sub_distrib_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "sub_distrib t = (\<forall>P s. sound P \<longrightarrow> t P s \<ominus> 1 \<le> t (\<lambda>s'. P s' \<ominus> 1) s)"
INF_eq_minf: fixes f :: "'a \<Rightarrow> ereal"   and I :: "'a set" shows "(Inf (f ` I) \<noteq> - \<infinity>) = (\<exists>b>- \<infinity>. \<forall>i\<in>I. b \<le> f i)"
eng_mult: fixes h1 :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and h2 :: "'a" assumes "h1 \<in> generate_ring R H"   and "h2 \<in> generate_ring R H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> generate_ring R H"
assms: fixes r :: "'a list"   and s :: "'a list" shows "\<forall>x\<in>set xs. showx x (r @ s) = showx x r @ s"
Shuffle_def: fixes r :: "'b language"   and s :: "'b language" shows "Shuffle r s = Times_Plus (ShuffleLR r s)"
I_Suc: fixes n :: "nat" shows "I (Suc n) = real (2 * (2 * n + 1)) / real (n + 2) * I n"
restrict_PiE: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
d_indistinguishable_sym: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and e :: "'a"   and e' :: "'a" assumes "WHATWHERE_Secure_Programs.d_indistinguishable E DA d e e'" shows "WHATWHERE_Secure_Programs.d_indistinguishable E DA d e' e"
hyps_2: fixes W :: "char list set" assumes "fv e \<subseteq> W" shows "P \<turnstile> \<langle>e,(h, l |` W, sh)\<rangle> \<Rightarrow> \<langle>throw a,(h', l' |` W, sh')\<rangle>"
is_rat_real_def: fixes x :: "real" shows "is_rat x = (x \<in> \<rat>)"
word_1_le_power: fixes n :: "nat" assumes "n < LENGTH('b)" shows "1 \<le> 2 ^ n"
conc_Diff_if_Nil1: fixes A :: "'a list set"   and B :: "'a list set" assumes "[] \<in> A" shows "A @@ B = (A - {[]}) @@ B \<union> B"
similarI: fixes x :: "Value"   and y :: "CValue" assumes "\<And>n. \<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>y" shows "x \<triangleleft>\<triangleright> y"
preserves_arr: fixes f :: "'a" assumes "B.arr f" shows "B.arr (B.R f)"
tdghm_dghm_comp_components_1: fixes \<NN> :: "V"   and \<HH> :: "V" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>)\<lparr>NTMap\<rparr> = (\<lambda>a\<in>\<^sub>\<circ>\<HH>\<lparr>HomDom\<rparr>\<lparr>Obj\<rparr>. \<NN>\<lparr>NTMap\<rparr>\<lparr>\<HH>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr>)"
newton_seq_fact2: fixes k :: "nat" shows "eint 2 * val_Zp (f' \<bullet> a) + eint (2 ^ k) * t \<le> val_Zp (f \<bullet> ns k)"
slice_kinds_def: fixes S :: "'node SDG_node set"   and as :: "'edge list" shows "slice_kinds S as \<equiv> map (slice_kind S) as"
ge_eq_refl: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "(=) \<le> R" shows "R x x"
left_total_rel_pred: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique A" shows "left_total (rel_pred A)"
Un_empty_left: fixes B :: "'f set" shows "{} \<union> B = B"
cnf_singleton_false: fixes as :: "'b negation_type list"   and \<gamma> :: "'b \<Rightarrow> bool" shows "(\<exists>a'\<in>set as. \<not> cnf_to_bool \<gamma> [a']) = (\<not> cnf_to_bool \<gamma> as)"
disjunct_not_in: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A \<inter># B = {#}) = (\<forall>a. a \<notin># A \<or> a \<notin># B)"
src_hcomp: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.src \<nu> = Span.trg \<mu>" shows "Span.src (\<nu> \<circ> \<mu>) = Span.src \<mu>"
Conseq: fixes P' :: "state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and Q' :: "state \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile> {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile> {P'} c {Q'}"
path_image_part_circlepath': fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "path_image (part_circlepath z r s t) = (\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t"
drop_suffix_same_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and xs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and t :: "('e, 'f) xstate"   and xss :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs @ xs, css, s) \<rightarrow> (cs' @ xs, css, t)"   and "cs \<noteq> []" shows "\<Gamma>\<turnstile> (cs, xss, s) \<rightarrow> (cs', xss, t)"
freshAbs_psubstAbs_updEnv: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodAbs A"   and "goodEnv rho"   and "freshAbs xs x A" shows "A $[rho [x \<leftarrow> X]_xs] = A $[rho]"
as_needed_asses_submap_exec_xii': fixes v :: "'d"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "v \<notin> fmdom' f"   and "v \<in> fmdom' g" shows "fmlookup (f ++ g) v = fmlookup g v"
backlinks_set: fixes E :: "('a \<times> 'a) list" shows "set (backlinks E) = {(e2, e1). (e1, e2) \<in> set E}"
last_fin_make_untimed_inf_truncate: fixes z :: "nat \<Rightarrow> 'a list"   and i :: "nat"   and a :: "'a" assumes "z i = [a]" shows "last (fin_make_untimed (inf_truncate z i)) = a"
l2wi_remdups: fixes ls :: "('b word \<times> 'b word) list" shows "wordinterval_to_set (l2wi (remdups ls)) = wordinterval_to_set (l2wi ls)"
classical_operator_adjoint: fixes \<pi> :: "'c \<Rightarrow> 'd option" assumes "inj_map \<pi>" shows "classical_operator \<pi>* = classical_operator (inv_map \<pi>)"
Span_eq_generate: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "Span K Us = generate (add_monoid R) (K <#> set Us)"
single_vote_to_RPR_iff: fixes a :: "'c"   and A :: "'c set"   and x :: "'c"   and b :: "'c" assumes "a \<in> A"   and "x \<in> A"   and "a \<noteq> x" shows "(a \<^bsub>(single_vote_to_RPR A b)\<^esub>\<prec> x) = (b = a)"
trace_cons: fixes conf' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and final :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and conf :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "conf' \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> final"   and "conf \<Rightarrow> conf'" shows "conf \<Rightarrow>\<^sup>*\<^bsub>conf' # T\<^esub> final"
cong': fixes A :: "'a vec"   and B :: "'a vec"   and C :: "'a vec set" assumes "A = B"   and "A \<in> C" shows "B \<in> C"
map_ident: fixes t :: "'b error_result" shows "map_error_result (\<lambda>x. x) t = t"
abel_semigroup_ow_axioms: shows "abel_semigroup_ow U (*\<^sub>o\<^sub>w)"
d: shows "n < stp \<and> abc_steps_l (0, args @ 0 # 0 \<up> (c - Suc (length args)) @ anything) (a @ B) stp = (0, args @ Suc n # 0 \<up> (c - Suc (length args)) @ anything)"
simps_13: fixes n :: "nat"   and p :: "fm" shows "boundslt n (A p) = boundslt (Suc n) p"
intros_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> - b) c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c\<^sub>1 c\<^sub>2 Q,A"
inf_compl_bot: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"
bnds_sqrt: fixes l :: "float"   and u :: "float"   and prec :: "nat" shows "\<forall>x lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {real_of_float lx..real_of_float ux} \<longrightarrow> real_of_float l \<le> sqrt x \<and> sqrt x \<le> real_of_float u"
assms_1: shows "t \<in> GSMP (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>)"
recover_pure_aux: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" assumes "CONSTRAINT is_pure R" shows "hn_invalid R x y \<Longrightarrow>\<^sub>t hn_ctxt R x y"
earlyLateInput: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and x :: "name" assumes "P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'"   and "x \<sharp> P" shows "\<exists>P''. P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'' \<and> P' = P''[x::=u]"
in_Keys: fixes s :: "'c"   and F :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "(s \<in> Keys F) = (\<exists>f\<in>F. s \<in> keys f)"
\<pi>0': shows "\<pi>' 0 = \<pi> l"
refl_r: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "Semilat A r f"   and "x \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
primes_dvd_imp_eq: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "dvd.dvd times p q" shows "p = q"
octo_mult_scaleR_left: fixes r :: "real"   and x :: "octo"   and y :: "octo" shows "r *\<^sub>R x * y = r *\<^sub>R (x * y)"
l11_44_2_a: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C"   and "B A Lt B C" shows "B C A LtA B A C"
objE: fixes a :: "'d" assumes "D.obj a" obtains "D.ide a" and "src\<^sub>D a = a" and "trg\<^sub>D a = a"
has_derivative_at: fixes f :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "(f has_derivative D) (at x) = (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0::'c\<rightarrow> 0)"
list_splits_rev: fixes xs :: "'b list" shows "list_splits (rev xs) = map (\<lambda>(xs, x, ys). (rev ys, x, rev xs)) (rev (list_splits xs))"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c" shows "Null \<noteq> MkArr x21 x22 x23"
d: fixes f :: "'a ratfps" assumes "f \<in> A" shows "d dvd f"
a_1: shows "real_of_float a \<le> f' (real_of_float l)"
nths_append: fixes l :: "'c list"   and l' :: "'c list"   and A :: "nat set" shows "nths (l @ l') A = nths l A @ nths l' {j. j + length l \<in> A}"
False: shows "fst instr \<noteq> sreg_type WRY"
reduced_GB_is_reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.is_reduced_GB (punit.reduced_GB F)"
g_iso: shows "\<forall>x y. x \<le> y \<longrightarrow> g x \<le> g y"
1: shows "{a. ren a q} \<subset> {a. en a q}"
pred_mono_strong: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a bdd"   and Pa :: "'a \<Rightarrow> bool" assumes "bdd_all P x"   and "\<And>z. \<lbrakk>z \<in> set_bdd x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "bdd_all Pa x"
Chernoff_on_iia_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "Chernoff_on A f = iia_on A f"
exists_unique_long_division: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []" shows "\<exists>!t. long_divides p q t"
keyfree_CryptE: fixes K :: "nat"   and X :: "msg" assumes "Crypt K X \<in> keyfree" obtains "X \<in> keyfree"
vec_to_iarray_def: fixes A :: "('a, 'b) vec" shows "vec_to_iarray A = IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat i) CARD('b)"
continuous_at_avoid: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "isCont f x"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
strict_coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "c < b" shows "c < sup a b"
inv_unique: fixes y :: "'a \<Rightarrow> 'a"   and x :: "'a \<Rightarrow> 'a"   and y' :: "'a \<Rightarrow> 'a" assumes "y \<circ> x = id"   and "x \<circ> y' = id" shows "y = y'"
irrefl: fixes a :: "'b DBMEntry" shows "\<not> a \<prec> a"
mult_mono_nonpos_nonneg: fixes a :: "'a"   and c :: "'a"   and d :: "'a"   and b :: "'a" assumes "a \<le> c"   and "d \<le> b"   and "a \<le> (0::'a)"   and "(0::'a) \<le> d" shows "a * b \<le> c * d"
inv_join: fixes l :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree"   and a :: "'a" assumes "inv l"   and "inv r" shows "inv (join l a r)"
select_convs_7: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cIntensityF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cIntensityF"
Meet_subset_antimono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "\<Sqinter>B \<sqsubseteq> \<Sqinter>A"
tanh_real_less_iff: fixes x :: "real"   and y :: "real" shows "(tanh x < tanh y) = (x < y)"
compat: shows "compatible_cs (\<Sum>(f, c)\<leftarrow>C. lec_of_constraint (f *R c)) (Le_Constraint Leq_Rel 0 d)"
simps_3: fixes P :: "('d, 'e, 'f) psi" shows "\<tau> \<prec> P = RTau P"
1: shows "y \<noteq> x \<and> qAFresh xs y X"
tendsto_poly: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and p :: "'c poly" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. poly p (f x)) \<longlongrightarrow> poly p a) F"
bpp_closed1: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and n :: "nat"   and y :: "'e"   and m :: "nat" assumes "assoc_bpp A f"   and "x \<in> add_set f A n"   and "y \<in> add_set f A m" shows "x \<^sub>f+ y \<in> add_set f A (n + m + Suc 0)"
assms_1: shows "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"
one_V_def: shows "1 \<equiv> ZFC_in_HOL.succ 0"
finite_chamber: fixes x :: "'a set" assumes "chamber x" shows "finite x"
leg_simps_3: shows "C.arr H\<nu>H\<pi>\<rho>.leg1"
openin_inject: fixes x :: "'c topology"   and y :: "'c topology" shows "(openin x = openin y) = (x = y)"
ord_spmf_mono: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'d spmf"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "ord_spmf A p q"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "ord_spmf B p q"
sqrtl_4_eq: fixes x :: "real" assumes "0 \<le> x" shows "sqrtl x 4 = 32 * x * (x + 1) * (x\<^sup>2 + 6 * x + 1) * (x ^ 4 + 28 * x ^ 3 + 70 * x\<^sup>2 + 28 * x + 1) * (1820 * x ^ 6 + 8008 * x ^ 5 + x ^ 8 + 120 * x ^ 7 + 120 * x + 1 + 1820 * x\<^sup>2 + 8008 * x ^ 3 + 12870 * x ^ 4) / (225792840 * x ^ 6 + 64512240 * x ^ 5 + 601080390 * x ^ 8 + 471435600 * x ^ 7 + 496 * x + 1 + 35960 * x\<^sup>2 + 906192 * x ^ 3 + 10518300 * x ^ 4 + x ^ 16 + 906192 * x ^ 13 + 10518300 * x ^ 12 + 496 * x ^ 15 + 35960 * x ^ 14 + 471435600 * x ^ 9 + 225792840 * x ^ 10 + 64512240 * x ^ 11)"
multiset_induct: fixes P :: "'a multiset \<Rightarrow> bool"   and M :: "'a multiset" assumes "P {#}"   and "\<And>x M. P M \<Longrightarrow> P (add_mset x M)" shows "P M"
map_update: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and m :: "'a"   and a :: "'b"   and b :: "'c" assumes "Map empty update delete lookup invar"   and "invar m" shows "lookup (update a b m) = lookup m(a \<mapsto> b)"
Init_simps2_5: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "Init (P \<or> Q) \<equiv> Init P \<or> Init Q"
full_multiset_induct: fixes P :: "'b multiset \<Rightarrow> bool"   and B :: "'b multiset" assumes "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" shows "P B"
dhops_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "ip \<noteq> dip" shows "dhops (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = dhops rt ip"
R: fixes x :: "'a" assumes "x \<cdot> x = x" shows "x\<^sup>\<star> = (1::'a) + x"
assms_1: fixes x :: "'a" shows "spmf p x \<le> spmf q x"
fmsubset_restrict_set_mono: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap"   and A :: "'c set" assumes "m \<subseteq>\<^sub>f n" shows "fmrestrict_set A m \<subseteq>\<^sub>f fmrestrict_set A n"
IH_3: fixes ys :: "[:'a:]" shows "prefix\<cdot>xs\<cdot>ys = eq\<cdot>xs\<cdot>(stake\<cdot>(slength\<cdot>xs)\<cdot>ys)"
apex_def: shows "\<mu>.dom.apex \<equiv> C.dom \<mu>.dom.leg0"
elts_VPow: fixes x :: "V" shows "elts (VPow x) = set ` Pow (elts x)"
HEndPhase0_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4a s q"   and "HInv4d s p"   and "Inv2a s"   and "Inv1 s"   and "Inv2a s"   and "Inv2c s" shows "HInv4a s' q"
assms: shows "list_all2 (\<le>\<^sub>\<complex>) xs ys"
Rep_inverse: fixes x :: "'b word" shows "word_of_int (uint x) = x"
snd_map_prod: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'd"   and x :: "'f \<times> 'g" shows "snd (map_prod f g x) = g (snd x)"
rel_FGf_neg_distr_cond_def: fixes Co1 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Co1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Contra1 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Contra1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and uu :: "('y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'ak) itself" shows "rel_FGf_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1' L2 L2'. rel_FGf (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 OO rel_FGf L1' L2' Co1' Co2' Contra1' Contra2')"
sym: fixes x :: "'a"   and y :: "'a" assumes "x .= y"   and "x \<in> carrier S"   and "y \<in> carrier S" shows "y .= x"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq> z"
na2da_def: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)" shows "na2da A = ({start A}, \<lambda>a Q. \<Union> (next A a ` Q), \<lambda>Q. \<exists>q\<in>Q. fin A q)"
hom_inv: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id (inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x) = inv\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub> hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id x"
field_cont_on_path: shows "continuous_on ((\<lambda>x. \<gamma> x \<bullet> i) ` {0..1}) (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
index_map_mat_2: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d mat" shows "dim_row (map_mat f A) = dim_row A"
Max_wrt_preorder: shows "Max_wrt le = {x \<in> carrier. \<forall>y\<in>carrier. le x y \<longrightarrow> le y x}"
f_def: shows "f \<equiv> \<lambda>x. SUP i. MAX n\<in>{..i}. gs n x"
bounded_linear_right: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d" assumes "bounded_bilinear prod" shows "bounded_linear (prod a)"
simps_2: fixes mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a" shows "bal \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = bal"
term_cases: fixes P :: "Term.term \<Rightarrow> bool"   and t :: "Term.term" assumes "\<And>v. P (In2 v)"   and "\<And>e. P (In1l e)"   and "\<And>c. P (In1r c)"   and "\<And>l. P (In3 l)" shows "P t"
HSem_below: fixes h :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value"   and r :: "'var \<Rightarrow> 'value" assumes "\<And>x. x \<notin> domA h \<Longrightarrow> \<rho> x \<sqsubseteq> r x"   and "\<And>x. x \<in> domA h \<Longrightarrow> \<lbrakk> the (map_of h x) \<rbrakk>\<^bsub>r\<^esub> \<sqsubseteq> r x" shows "\<lbrace> h \<rbrace>\<rho> \<sqsubseteq> r"
nat_term_compD1': fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "comparator cmp"   and "nat_term_comp cmp"   and "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"   and "fst (rep_nat_term u) = 0" shows "cmp u v \<noteq> Gt"
ord_option_map2: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and f :: "'c \<Rightarrow> 'b"   and y :: "'c option" shows "ord_option ord x (map_option f y) = ord_option (\<lambda>x y. ord x (f y)) x y"
locally_compact_Int: fixes S :: "'b set"   and t :: "'b set" assumes "locally compact S"   and "locally compact t" shows "locally compact (S \<inter> t)"
degree_monom0: fixes a :: "'b" shows "degree (monom a 0) = 0"
expandComplete: fixes F :: "pi set" assumes "finite F" shows "\<exists>P. (P, F) \<in> sumComposeSet"
lemScaleLinear: fixes p :: "'a Point"   and a :: "'a"   and u :: "'a Vector"   and b :: "'a"   and v :: "'a Vector" shows "p \<leadsto> a ** u \<leadsto> b ** v = p \<leadsto> (a ** u \<oplus> b ** v)"
vars_subterms_of: fixes s :: "'b trm"   and t :: "'b trm" assumes "s \<in> subterms_of t" shows "vars_of s \<subseteq> vars_of t"