hcju/isabelleps huggingface dataset:
interval_eq_set_of_iff: fixes X :: "'a interval"   and Y :: "'a interval" shows "(X = Y) = (set_of X = set_of Y)"
sums_Dirichlet_L: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>n. \<chi> n * of_nat n powr - s) sums Dirichlet_L n \<chi> s"
intros: fixes q :: "'a"   and f :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set"   and ts :: "'b tree list" assumes "q \<rightarrow> f qs \<in> \<delta>"   and "list_all_zip (accs_laz \<delta>) ts qs" shows "accs_laz \<delta> (NODE f ts) q"
simps_1: shows "C 0 = pderiv p div gcd p (pderiv p)"
step_3_c_inner_loop_similar: fixes A :: "'b mat"   and n :: "nat"   and l :: "nat"   and i :: "nat"   and k :: "nat"   and val :: "'b" assumes "A \<in> carrier_mat n n"   and "l \<noteq> i"   and "k - 1 \<le> l"   and "k - 1 \<le> i"   and "l < n"   and "i < n" shows "similar_mat (step_3_c_inner_loop val l i k A) A"
np: shows "\<not> {} \<turnstile> \<delta> \<and> \<not> {} \<turnstile> Neg \<delta>"
dlex_pm_strict_alt: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "dlex_pm_strict s t = (\<not> dlex_pm t s)"
Span_subalgebra_minimal: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "Span K Us = \<Inter> {V. subalgebra K V R \<and> set Us \<subseteq> V}"
list_all2_append1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list"   and zs :: "'h list" shows "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)"
simps_14: fixes v :: "fm"   and va :: "fm" shows "conjuncts (Imp v va) = [Imp v va]"
power_increasing: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and n :: "nat"   and N :: "nat"   and a :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq one a" shows "less_eq (power.power one times a n) (power.power one times a N)"
has_derivative_cos: fixes g :: "'c \<Rightarrow> real"   and g' :: "'c \<Rightarrow> real"   and x :: "'c"   and s :: "'c set" assumes "(g has_derivative g') (at x within s)" shows "((\<lambda>x. cos (g x)) has_derivative (\<lambda>xa. g' xa * - sin (g x))) (at x within s)"
b: shows "k < length cms\<^sub>1 \<and> (cms\<^sub>1 ! k, mem\<^sub>1) \<leadsto> (cms\<^sub>1' ! k, mem\<^sub>1') \<and> cms\<^sub>1' = cms\<^sub>1[k := cms\<^sub>1' ! k]"
in_carrier_2: shows "local.eval q' a \<in> carrier R"
foundation21: fixes A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(not A \<triangleq> not B) = (A \<triangleq> B)"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier \<G>" shows "x [^] n \<in> carrier \<G>"
iMOD_div_right_strict_mono_on: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "k \<le> m" shows "SetInterval2.strict_mono_on (\<lambda>x. x div k) [ r, mod m ]"
coboundedI1: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<le> a" shows "c \<le> sup a b"
inf_left_commute: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
ereal_less_uminus_reorder: fixes a :: "ereal"   and b :: "ereal" shows "(a < - b) = (b < - a)"
in_snapshot_def: fixes r :: "'d" shows "in_snapshot r = (\<lambda>s. black r s \<or> (\<exists>x. (x grey_protects_white r) s))"
assms_4: shows "\<forall>k\<in>set (i # j # vs). 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)"
st: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "sound (t P)"
simps_3: fixes c :: "'e" shows "rsuml (Inr c) = Inr (Inr c)"
observable_io_targets_suffix: fixes M :: "('d, 'e, 'f) FSM"   and q1 :: "'f"   and vs :: "('d \<times> 'e) list"   and q2 :: "'f"   and xs :: "('d \<times> 'e) list"   and q3 :: "'f" assumes "observable M"   and "io_targets M q1 vs = {q2}"   and "io_targets M q1 (vs @ xs) = {q3}" shows "io_targets M q2 xs = {q3}"
logical_symmetry: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s is t : T" shows "\<Gamma> \<turnstile> t is s : T"
suspendable_formula_simp_3: fixes \<xi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "suspendable \<xi>" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<xi> = w \<Turnstile>\<^sub>n \<xi>"
expands_to_imp_eventually_nz: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "trimmed F" shows "\<forall>\<^sub>F x in at_top. f x \<noteq> 0"
lookup_inject: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and y :: "'b \<Rightarrow>\<^sub>0 'c" shows "(poly_mapping.lookup x = poly_mapping.lookup y) = (x = y)"
case_optionE: fixes P :: "bool"   and Q :: "'g \<Rightarrow> bool"   and x :: "'g option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'g" where "x = Some y" and "Q y"
large: shows "2 ^ (5 * (degree F - 1) * (degree F - 1)) * \<parallel>F\<parallel>\<^sup>2 ^ (2 * (degree F - 1)) < pl\<^sup>2"
min_int_poly_conv_Gcd: fixes x :: "'a" assumes "algebraic x" shows "min_int_poly x = Gcd {p. p \<noteq> 0 \<and> p represents x}"
Hausdorff_space_def: fixes X :: "'c topology" shows "Hausdorff_space X \<equiv> \<forall>x y. x \<in> topspace X \<and> y \<in> topspace X \<and> x \<noteq> y \<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V)"
isometry_inverse_1: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "isometry (inv f)"
totient_primorial'_asymp_equiv': shows "(\<lambda>x. real (totient (primorial' x))) \<sim>[sequentially] (\<lambda>k. third_mertens_const * real (primorial' k) / ln (ln (real (primorial' k))))"
true_cls_empty: fixes I :: "'b set" shows "\<not> I \<Turnstile> {#}"
soundness_diagonalize_JNF: fixes m :: "nat"   and n :: "nat" shows "\<forall>A bezout. A \<in> carrier_mat m n \<and> is_bezout_ext bezout \<longrightarrow> (case diagonalize_JNF A bezout of (P, S, Q) \<Rightarrow> P \<in> carrier_mat m m \<and> Q \<in> carrier_mat n n \<and> S \<in> carrier_mat m n \<and> invertible_mat P \<and> invertible_mat Q \<and> isDiagonal_mat S \<and> S = P * A * Q)"
SPRAY_def: fixes x :: "'a" shows "SPRAY x \<equiv> {R \<in> \<P>. x \<in> R}"
RPDs_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Past Strict (from_mregex mr \<phi>s)"   and "ms \<in> RPDs mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
arev_arev_raw: fixes a :: "'b" assumes "a \<in> arcs G" shows "arev (arev a) = a"
parallel_composition: fixes V :: "('exp, 'id) MWLsCom list" assumes "\<forall>i<length V. WHATWHERE_Secure [V ! i]"   and "unique_PPV V" shows "WHATWHERE_Secure V"
measure_eq_emeasure_eq_ennreal: fixes x :: "real"   and M :: "'c measure"   and A :: "'c set" assumes "0 \<le> x"   and "emeasure M A = ennreal x" shows "Sigma_Algebra.measure M A = x"
simps_2: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "x # xs \<simeq> {y # ys, [], P} = (x = y \<and> xs \<simeq> {ys, [], P})"
If_not: fixes a :: "bool"   and b :: "'b"   and c :: "'b" shows "(if \<not> a then b else c) = (if a then c else b)"
is_mgu_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) 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>))"
exp_inv_limit_0_right_gen': fixes k :: "nat" shows "((\<lambda>t. inverse (t ^ k) / exp (inverse t)) \<longlongrightarrow> 0) (at_right 0)"
sym_iff: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "sympartp r x y = sympartp r y x"
lm107: fixes X :: "'b set"   and Z :: "'b set" shows "sum (chi X) (Z - X) = (0::'a)"
prefix_append: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
stateT_abs_def: shows "stateT_abs \<equiv> PRJ('b\<cdot>('c, 'a) stateT) oo EMB('a \<rightarrow> ('b \<times> 'a)\<cdot>'c)"
hyps_2: fixes cs :: "('b, 'a, 'c) com list"   and css :: "(('b, 'a, 'c) com list \<times> ('b, 'a, 'c) com list) list" shows "\<Gamma>\<turnstile> (c\<^sub>1 # cs, css, Normal s) \<rightarrow>\<^sup>* (cs, css, s')"
rel_mset_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'d multiset"   and Y :: "'e multiset" shows "rel_mset R X Y = (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
supp_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
case_1: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and x1 :: "'c" shows "(case bot.Value x1 of bot.Value x \<Rightarrow> f1 x | bot.Bot \<Rightarrow> f2) = f1 x1"
drop_all: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
cons1: fixes env :: "v sem_env"   and st :: "'b state"   and d :: "dec"   and mn :: "char list list"   and ds :: "dec list" assumes "dec_diverges env st d" shows "decs_diverges mn env st (d # ds)"
is_acc_def: fixes G :: "('a, 'b) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "gb_graph G" shows "gb_graph.is_acc G r \<equiv> \<forall>A\<in>gbg_F G. \<exists>\<^sub>\<infinity>i. r i \<in> A"
cases: fixes x :: "'b list \<times> 'b list \<times> 'b list" obtains x :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" and z :: "'b" and zs :: "'b list" where "xa__ = (xb__ # xsa__, y__ # ys__, z # zs__)"   | x :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" where "xa__ = (x # xs, y # ys, [])"   | uv_ :: "'b list" and zs :: "'b list" where "xa__ = ([], uv___, zsa__)"   | uu_ :: "'b list" and zs :: "'b list" where "xa__ = (uu___, [], zs)"   | uv_ :: "'b list" where "xa__ = ([], uv_, [])"   | uu_ :: "'b list" where "xa__ = (uu_, [], [])"
oldBlks: shows "bk \<in> blocksOf s q \<longrightarrow> Inv2a_innermost s q bk"
dis_rews_2: shows "is_EQ\<cdot>\<bottom> = \<bottom>"
r: shows "r = q g1 \<odot> g1 + q g2 \<odot> g2 + (\<Sum>g\<in>G - {g1, g2}. q g \<odot> g)"
SSpec_strong: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes " \<Turnstile> c : A " shows "\<forall>s t. SSpec c s t \<longrightarrow> A s t"
element_ptr_kinds_eq_h3: shows "element_ptr_kinds h' = element_ptr_kinds h3"
red2_Nml: fixes a :: "'c local.term"   and b :: "'c local.term" assumes "Nml (a \<^bold>\<star> b)" shows "a \<^bold>\<Down> b = a \<^bold>\<star> b"
r: shows "{x. poly (complex_of_int_poly p) x = 0} = {x. poly (map_poly complex_of_real q) x = 0}"
add_left_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "c + a \<le> c + b"
of_nat_in_Nats: fixes n :: "nat" shows "of_nat n \<in> \<nat>"
\<phi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<phi> (b, a) \<in> C.hom b a \<rightarrow> local.set (b, a)"
simps: fixes f :: "nat \<Rightarrow> 'a"   and j :: "nat" shows "shift f j = (\<lambda>i. f (i + j))"
llist_all2_conv_all_lnth: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>n. enat n < llength ys \<longrightarrow> P (lnth xs n) (lnth ys n)))"
PrefixListMonotonicity: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "prefixList l1 l2" shows "length l1 < length l2"
inverts_components: fixes a :: "'a \<times> 'a \<times> 'a" assumes "CCC.ide a" shows "inverse_arrows (\<alpha> a) (\<alpha>' a)"
finite_Psegments_Re_imp_jumpFs: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and a :: "real"   and b :: "real" assumes "finite_Psegments (\<lambda>t. Re (g t - z) = 0) a b"   and "continuous_on {a..b} g" shows "finite_jumpFs (\<lambda>t. Im (g t - z) / Re (g t - z)) a b"
annul_val_def: fixes state :: "'b sparc_state" shows "annul_val state \<equiv> get_annul (state_var state)"
continuous_on_open_UN: fixes S :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>s. s \<in> S \<Longrightarrow> open (A s)"   and "\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f" shows "continuous_on (\<Union> (A ` S)) f"
lsu_iso_var: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "\<nu>\<^sup>\<natural> x \<le> \<nu>\<^sup>\<natural> y"
assms_2: shows "a \<in>\<^sub>\<circ> set {f x y |x y. P x y}"
subgame_E: fixes V' :: "'a set" shows "E\<^bsub>G'.subgame V'\<^esub> \<subseteq> E\<^bsub>subgame V'\<^esub>"
homeomorphic_convex_compact: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "compact S"   and "interior S \<noteq> {}"   and "convex T"   and "compact T"   and "interior T \<noteq> {}" shows "S homeomorphic T"
sum_fun_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite A" shows "\<Sum> ((\<lambda>x. f x + g x) ` A) = \<Sum> (f ` A) + \<Sum> (g ` A)"
Lcm_insert: fixes a :: "'a"   and A :: "'a set" shows "Lcm (insert a A) = lcm a (Lcm A)"
hyps_4: fixes b :: "(name \<times> ty) list" assumes "valid b"   and "(x, T1) # \<Gamma>1 \<subseteq> b" shows "b \<turnstile> t : T2"
simps_3: fixes F :: "'b formula"   and G :: "'b formula" shows "is_nnf (F \<^bold>\<or> G) = (is_nnf F \<and> is_nnf G)"
netmap_fst_netgmap_rel: fixes sr :: "'b \<Rightarrow> 'a \<times> 'c"   and s :: "'b net_state" shows "(\<lambda>i. map_option (fst \<circ> sr) (netmap s i)) = fst (netgmap sr s)"
segment_segment: fixes b :: "'a"   and a :: "'a" assumes "b \<in> carrier D"   and "a \<in> segment D b" shows "segment (Iod D (segment D b)) a = segment D a"
lnull_ldrop: fixes n :: "enat"   and xs :: "'c llist" shows "lnull (ldrop n xs) = (llength xs \<le> n)"
s_prod_idl: fixes x :: "('a \<times> 'a set) set" shows "1\<^sub>\<sigma> \<cdot> x = x"
R: shows "\<forall>i\<in>I. subgroup (H i) (G i)"
h1: shows "finite A \<and> maximal A (\<lambda>S. S \<subseteq> carrier V \<and> lin_indpt S)"
assms: shows "(\<lambda>x y. R x y \<and> P x y)\<^sup>+\<^sup>+ x y"
isolated_zeros: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex"   and \<beta> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "\<xi> \<in> S"   and "f \<xi> = 0"   and "\<beta> \<in> S"   and "f \<beta> \<noteq> 0" obtains r :: "real" where "0 < r" and "ball \<xi> r \<subseteq> S" and "\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
COD_comp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "ARR f"   and "ARR g"   and "DOM f = COD g" shows "COD (f \<cdot> g) = COD f"
intros_5: fixes c :: "('d, 'e, 'f) com"   and b :: "'d set" shows "com_relp c (While b c)"
F2_o_F1_mono: fixes ds :: "'b set" shows "mono (F2 ds \<circ> F1)"
simps_1: fixes a :: "int"   and p :: "int point" shows "point_mult a 0 p = Infinity"
lookup_not_edom: fixes x :: "'b"   and m :: "'b \<Rightarrow> 'c" assumes "x \<notin> edom m" shows "m x = \<bottom>"
AE_pmfI: fixes M :: "'c pmf"   and P :: "'c \<Rightarrow> bool" assumes "\<And>y. y \<in> set_pmf M \<Longrightarrow> P y" shows "almost_everywhere (measure_pmf M) P"
select_convs_1: fixes committed :: "nat set"   and justifying_exec :: "('b \<times> ('a, 'b) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and more :: "'c" shows "committed \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, \<dots> = more\<rparr> = committed"
Cons: fixes s' :: "('St, 'U, 'C) state"   and s :: "('St, 'U, 'C) state"   and sl :: "('St, 'U, 'C) state list" assumes "s' \<in> \<delta> s"   and "wffp (s' # sl)" shows "wffp (s # s' # sl)"
add_additive_subgroups: fixes H :: "'a set"   and K :: "'a set" assumes "additive_subgroup H R"   and "additive_subgroup K R" shows "additive_subgroup (H <+> K) R"
card_set_mset_leq: fixes M :: "'c multiset" shows "card (set_mset M) \<le> size M"
assms_3: shows "NTDom \<eta>1 = NTDom \<eta>2"
count: fixes x :: "('c \<times> nat) list" shows "count (Abs_multiset (count_of x)) = count_of x"
converse_rtranclE: fixes x :: "'x"   and z :: "'x"   and r :: "('x \<times> 'x) set" assumes "(x, z) \<in> r\<^sup>*" obtains "x = z"   | y :: "'x" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>*"
cinner_divide_left: fixes a :: "'a"   and m :: "complex"   and b :: "'a" shows "cinner (a / of_complex m) b = cinner a b / cnj m"
image_mset_is_empty_iff: fixes f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset" shows "(image_mset f M = {#}) = (M = {#})"
leq_refl: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set"   and x :: "'c" assumes "class.stimuli leq add seq_comp neutral deactivation basic" shows "leq x x"
intros_1: fixes u :: "'e \<Rightarrow> 'd"   and r :: "'e"   and d :: "'d" assumes "u r \<le> d" shows "dbm_entry_val u (Some r) None (Le d)"
case_prodD: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and b :: "'c" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
obtain_const_tm: fixes e :: "(name, hf) finfun"   and x :: "hf" obtains t :: "tm" where "\<lbrakk>t\<rbrakk>e = x" and "ground t"
len: fixes n :: "nat" shows "length (f n) \<le> m'"
assms_4: shows "(p, ins, outs) \<in> set procs"
prime_ideal_ideal: fixes I :: "'a set" assumes "prime_ideal R I" shows "ideal R I"
coeff_bound: shows "fds_nth F \<in> O(\<lambda>_. 1)"
cSUP_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and c :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}" shows "Sup ((\<lambda>x. c) ` A) = c"
P13_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b r\<rightarrow> c \<le> a r\<rightarrow> c"
wf_syscls_xcpt_subcls_Throwable: fixes C :: "String.literal"   and P :: "'a prog" assumes "C \<in> sys_xcpts"   and "wf_syscls P" shows "P \<turnstile> C \<preceq>\<^sup>* Throwable"
fermat_witness_def: fixes a :: "nat"   and n :: "nat" shows "fermat_witness a n = [a ^ (n - 1) \<noteq> 1] (mod n)"
FVar: fixes G :: "prog"   and s0 :: "st"   and statDeclC :: "qtname"   and n :: "nat"   and s1 :: "state"   and e :: "expr"   and a :: "val"   and s2 :: "state"   and v :: "vvar"   and s2' :: "state"   and stat :: "bool"   and fn :: "vname"   and s3 :: "state"   and accC :: "qtname" assumes "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s2"   and "(v, s2') = fvar statDeclC stat fn a s2"   and "s3 = check_field_access G accC statDeclC fn stat a s2'" shows "G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<midarrow>n\<rightarrow> s3"
ide_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxVxV.ide f = (ide (fst f) \<and> VxV.ide (snd f))"
add_divide_eq_if_simps_2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and z :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus (divide a z) b = (if z = zero then b else divide (plus a (times b z)) z)"
return_env_def: fixes return :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "return_env return x = EnvT (\<lambda>_. return x)"
upto_split1: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "j \<le> k" shows "[i..k] = [i..j - 1] @ [j..k]"
iface_conjunct: fixes i1 :: "iface"   and p_i :: "char list"   and i2 :: "iface" shows "(match_iface i1 p_i \<and> match_iface i2 p_i) = (case iface_conjunct i1 i2 of None \<Rightarrow> False | Some x \<Rightarrow> match_iface x p_i)"
imirror_bounds_iMin: fixes I :: "nat set"   and l :: "nat"   and r :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "iMin I \<le> l + r" shows "iMin (imirror_bounds I l r) = l + r - Max I"
simps_1: fixes l :: "nat list" shows "levelsCounter_aux [] l = l"
cases: fixes x :: "('a, 'b, 'c) SLin_state \<times> ('a, 'b, 'c) SLin_state" obtains s1 :: "('a, 'b, 'c) SLin_state" and s2 :: "('a, 'b, 'c) SLin_state" where "x = (s1, s2)"
step: fixes l1 :: "'a list"   and x :: "'a"   and l2 :: "'a list"   and \<sigma> :: "'\<sigma>" assumes "l0 = l1 @ x # l2"   and "I \<sigma> l1 (x # l2)" shows "I (f \<sigma> x) (l1 @ [x]) l2"
gen_cong_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and j1 :: "'a"   and j2 :: "'a" shows "gen_cong R j1 j2 \<equiv> \<forall>R'. R \<le> R' \<and> local.cong R' \<longrightarrow> R' j1 j2"
is_ortho_set_cindependent: fixes A :: "'a set" assumes "is_ortho_set A" shows "cindependent A"
congruentD: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'c"   and z :: "'c" assumes "f respects r"   and "(y, z) \<in> r" shows "f y = f z"
less_minus_one_simps_2: 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" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus one) one"
outs'_gpv_lift_spmf: fixes p :: "'e spmf" shows "outs'_gpv (lift_spmf p) = {}"
exists_set_with_max_card: fixes S :: "'b set set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. card s = Max (card ` S)"
Done_Done: fixes a :: "'f option"   and a' :: "'f option" assumes "option_ord a a'" shows "resumption_ord (Resumption.resumption.Done a) (Resumption.resumption.Done a')"
invariant_1_pos_1: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)" shows "(0 < real_of_1 (p, l, r)) = (0 < r)"
sl_ys_g_ak: shows "a ! k < sum_list ys"
PR_def: shows "PR = prod_list (map ABP [3..<Suc i])"
isLub_lub: fixes S :: "'b set"   and cl :: "'b potype" shows "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
H_Inf_pres_fpower: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and i :: "nat" assumes "Inf_pres f"   and "x \<le> f x" shows "x \<le> fpower f i x"
comp2_def: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "'e"   and y :: "'f" shows "(f o2 g) x y \<equiv> f (g x y)"
simps_1: fixes u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and m :: "nat" shows "muh 0 u m = u m 0"
pure_eq: shows "CF_pure n \<leftrightarrow> CF_pure n'"