hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'a"   and a2 :: "'a list" assumes "ListMem a1 a2" obtains x :: "'a" and xs :: "'a list" where "a1 = x__" and "a2 = x__ # xs__"   | x :: "'a" and xs :: "'a list" and y :: "'a" where "a1 = x" and "a2 = y # xs" and "ListMem x xs"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'b" shows "(a *a x = (0::'b)) = (a = (0::'a) \<or> x = (0::'b))"
vec_last_def: fixes v :: "'a vec"   and n :: "nat" shows "vec_last v n \<equiv> vec n (\<lambda>i. v $ (dim_vec v - n + i))"
less_eqE: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subseteq># b" obtains c :: "'d multiset" where "b = a + c"
simple_fw_iff_generalized_fw: fixes fw :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme"   and a :: "simple_action" shows "(simple_fw fw p = simple_action_to_decision a) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, a))"
next_free_always: fixes \<phi> :: "'b pltl" shows "next_free (G\<^sub>p \<phi>) = next_free \<phi>"
inv_imp: fixes s :: "'S" assumes "s \<in> dwa_invar WA"   and "\<not> dwa_cond WA s" shows "Q s"
permChainFreshFresh_1: fixes xvec :: "name list"   and p :: "'a"   and yvec :: "name list" assumes "xvec \<sharp>* p"   and "yvec \<sharp>* p"   and "length xvec = length yvec" shows "[xvec yvec] \<bullet>\<^sub>v p = p"
blinfun_apply_snd_blinfun: shows "blinfun_apply snd_blinfun = snd"
flow_to_path_path: fixes t :: "real"   and t' :: "real"   and x :: "'a" assumes "t \<le> t'"   and "{t..t'} \<subseteq> existence_ivl0 x" shows "path (flow_to_path x t t')"
const_def: fixes x :: "'g" shows "pure x = (\<lambda>_. x)"
mapping_Assoc_List_def: shows "mapping_Assoc_List = Mapping_IMPL"
simps_3: fixes \<sigma> :: "'tp" shows "TE_wtFsym (Wit \<sigma>) = (\<not> isRes \<sigma>)"
c: shows "change_respecting (cms\<^sub>1 ! k, mem\<^sub>1) (cms\<^sub>1' ! k, mem\<^sub>1') (differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k) g1"
concat_eq_appendD: fixes xss :: "'g list list"   and ys :: "'g list"   and zs :: "'g list" assumes "concat xss = ys @ zs"   and "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2"
lipschitz_on_subset: fixes L :: "real"   and E :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and D :: "'b set" assumes "L-lipschitz_on E f"   and "D \<subseteq> E" shows "L-lipschitz_on D f"
fps_one_def: shows "1 = Abs_fps (\<lambda>n. if n = 0 then 1::'a else (0::'a))"
alpha_qOp_iff: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and delta' :: "'e"   and inp' :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp' :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "(X #= qOp delta' inp' binp') = (\<exists>inp binp. X = qOp delta' inp binp \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
execute_assert_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and h :: "heap" assumes "\<not> P x" shows "execute (assert P x) h = None"
reachable_def: shows "reachable \<equiv> \<lambda>F. {x. reachablep F x}"
extf_ext_irrefl: fixes f :: "'s" shows "ext_irrefl (extf f)"
comm_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "comm_group G \<equiv> Group.comm_monoid G \<and> Group.group G"
SETZFfun: fixes f :: "ZF"   and X :: "ZF"   and Y :: "ZF" assumes "f maps\<^bsub>SET\<^esub> X to Y" shows "f = ZFfun X Y ((|@|) f)"
ucast_le_ucast: fixes x :: "'b word"   and y :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)" shows "(UCAST('b \<rightarrow> 'c) x \<le> UCAST('b \<rightarrow> 'c) y) = (x \<le> y)"
lift_pref_profile_conv_vector: fixes i :: "'a"   and agents :: "'a set"   and agents' :: "'a set"   and alts :: "'b set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "i \<in> agents"   and "i \<in> agents'" shows "lift_pref_profile agents alts agents' alts' R i = lift_prefs alts alts' (R i)"
le_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "(m \<le> Suc n) = (m \<le> n \<or> m = Suc n)"
terminal_tllist_of_llist: fixes y :: "'a"   and xs :: "'b llist" shows "terminal (tllist_of_llist y xs) = (if lfinite xs then y else undefined)"
lcV: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "lead_coeff (#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.v' q_ty b u v w)) = @lead_coeff v"
map_ran_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "map_ran f = map (\<lambda>(k, v). (k, f k v))"
rule_WSC: fixes u :: "'a"   and xs :: "'b list"   and ys :: "'b list"   and P :: "'b process"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "(u, xs, ys) \<in> rel_induct_aux P I D"   and "(D x, xs, ys) \<in> rel_induct_aux P I D"   and "x \<in> next_events P xs \<inter> next_events P ys" shows "(u, xs @ [x], ys @ [x]) \<in> rel_induct_aux P I D"
subgraph_def: fixes G' :: "nat set \<times> nat set set"   and G :: "nat set \<times> nat set set" shows "subgraph G' G \<equiv> uverts G' \<subseteq> uverts G \<and> uedges G' \<subseteq> uedges G"
zero_minus_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n - v = - v"
foundation13: fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A \<triangleq> true) = (\<tau> \<Turnstile> A)"
prems_2: shows "\<not> occs (case v of (a, b) \<Rightarrow> Fv a b) (t1 $ t2)"
subgroup_conjugation_is_surj1: fixes g :: "'a"   and H :: "'a set" assumes "g \<in> carrier G"   and "subgroup H G" shows "subgroup (inv g <# H #> g) G"
unique_factorization_m_imp_factorization: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" assumes "unique_factorization_m f cfs" shows "factorization_m f cfs"
pcompose_const: fixes a :: "'b"   and q :: "'b poly" shows "[:a:] \<circ>\<^sub>p q = [:a:]"
inversion_number_between_permute: fixes \<pi>1 :: "nat \<Rightarrow> nat"   and xs :: "'a list"   and \<pi>2 :: "nat \<Rightarrow> nat"   and ys :: "'a list" assumes "\<pi>1 permutes {..<length xs}"   and "\<pi>2 permutes {..<length ys}" shows "inversion_number_between (permute_list \<pi>1 xs) (permute_list \<pi>2 ys) = inversion_number_between xs ys"
simps_3: fixes a :: "('a, 'b) Combinators"   and v :: "'a"   and va :: "'a" shows "member a (DenyAllFromTo v va) = (a = DenyAllFromTo v va)"
address_of_simps_2: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "address_of (Read\<^sub>s\<^sub>b volatile a t v) = {a}"
bernoulli_fps_aux: shows "bernoulli_fps * (fps_exp (1::'b) - 1) = fps_X"
simps_4: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "And p q[s/k] = And (p[s/k]) (q[s/k])"
distinct_mtf: fixes x :: "'a"   and xs :: "'a list" shows "distinct (mtf x xs) = distinct xs"
surjective: fixes r :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "r = \<lparr>orderA = \<X>\<^bsub>r\<^esub>, orderB = \<Y>\<^bsub>r\<^esub>, lower = \<pi>\<^sup>*\<^bsub>r\<^esub>, upper = \<pi>\<^sub>*\<^bsub>r\<^esub>, \<dots> = galcon.more r\<rparr>"
anycast_msg_zhops: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(uu_, a, uu_). anycast msg_zhops a)"
that: fixes \<T> :: "'a set set" assumes "simplicial_complex \<T>"   and "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e"   and "\<Union> \<T> = \<Union> \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F"   and "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C" shows "thesis"
semiring_simprules_8: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
moebius_ocircline_id_moebius: fixes H :: "ocircline" shows "moebius_ocircline id_moebius H = H"
const_term_explicit: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "p \<noteq> []"   and "const_term p = a" obtains p' :: "'a list" where "set p' \<subseteq> carrier R" and "p = p' @ [a]"
linorder_top_finite_sup: fixes f :: "'c \<Rightarrow> 'd" assumes "\<forall>k. f k \<noteq> top" shows "sup_monoid.sum f {k. True} \<noteq> top"
prime_dvd_other_side: fixes n :: "nat"   and p :: "nat"   and k :: "nat" assumes "n * n = p * (k * k)"   and "prime p" shows "p dvd n"
v: shows "v = term_of_pair (0::'a, component_of_term u)"
prems_1: fixes Q' :: "'state" assumes "Q \<Rightarrow> Q'"   and "P' \<approx>\<cdot> Q'" shows "thesis"
simple_extension_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.simple_extension (R\<lparr>carrier := K\<rparr>) = simple_extension"
normalize_1: 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" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize one = one"
filter_def: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c set" shows "Set.filter P A = rel_ext (\<lambda>a. a \<in> A \<and> P a)"
simps_20: fixes e :: "('f, 'g, 'h) exp"   and e\<^sub>1 :: "('f, 'g, 'h) exp"   and e\<^sub>2 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (if (e) e\<^sub>1 else e\<^sub>2) A = (\<D> e A \<and> \<D> e\<^sub>1 (A \<squnion> \<A> e) \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e))"
intros_44: fixes E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 "P,E \<turnstile> \<langle>throw null,s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
set_rel_witness_pmf: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf" assumes "rel_pmf A x y" shows "set_pmf (rel_witness_pmf A (x, y)) \<subseteq> {(a, b). A a b}"
Inf_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. less_eq a b" shows "less_eq (Inf A) (Inf B)"
domintros_4: fixes ux :: "char list"   and uy :: "exp0" shows "is_value_dom (Fun ux uy)"
comp_ipresIGFresh: fixes h :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'w, 'ae, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ab, 'ai, 'ad, 'x, 'aj, 'ak) model_scheme"   and h' :: "'x \<Rightarrow> 'al"   and MOD'' :: "('am, 'an, 'aa, 'ao, 'ap, 'ad, 'al, 'aq, 'ar) model_scheme" assumes "ipresIGWls h MOD MOD'"   and "ipresIGFresh h MOD MOD'"   and "ipresIGFresh h' MOD' MOD''" shows "ipresIGFresh (h' \<circ> h) MOD MOD''"
numeral_power_less_of_nat_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and i :: "num"   and n :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (power.power one times (numeral.numeral one plus i) n) (semiring_1.of_nat one plus zero x) = (numeral i ^ n < x)"
path_component: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "path_component S x y = (\<exists>t. path_connected t \<and> t \<subseteq> S \<and> x \<in> t \<and> y \<in> t)"
F_def: fixes index :: "nat \<times> nat"   and x :: "nat" shows "F index x = x ^ snd index mod fst index"
algebraic_times_i_1: fixes x :: "complex" assumes "algebraic x" shows "algebraic (\<i> * x)"
image_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and r :: "'c set" shows "\<P> f (\<P> g r) = \<P> (f \<circ> g) r"
restrict_inv_into_hom: fixes h :: "'a \<Rightarrow> 'a" assumes "h \<in> hom G G"   and "h \<in> Bij (carrier G)" shows "restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G"
wfzf_trancl: fixes R :: "(ZF \<times> ZF) set" assumes "wfzf R" shows "wfzf (R\<^sup>+)"
INTEG_id_eval: shows "UP_pre_univ_prop INTEG INTEG id"
distinct_21: fixes x21 :: "'c"   and x22 :: "'c \<Rightarrow> ('a, 'c) trm"   and x71 :: "'b"   and x72 :: "('a, 'b, 'c) formula" shows "$\<phi> x21 x22 \<noteq> InContext x71 x72"
funcset_mem1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and x :: "'c" assumes "\<forall>l\<in>A. f l \<in> B"   and "x \<in> A" shows "f x \<in> B"
fps_inverse_def: fixes f :: "'a fps" shows "inverse f = fps_right_inverse f (inverse (f $ 0))"
subcocycle_avg_ereal_def: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_avg_ereal u = \<Sqinter> {ereal (integral\<^sup>L M (u n) / real n) |n. 0 < n}"
rep_eq: fixes x :: "'c fset"   and xa :: "'c fset" shows "fset (x |-| xa) = fset x - fset xa"
False: shows "\<not> (k = 0 \<or> k = 1)"
ness: shows "\<not> essential (bipartite_web_of \<Gamma>) (B (bipartite_web_of \<Gamma>)) (TER\<^bsub>bipartite_web_of \<Gamma>\<^esub> f) (Inl a')"
map_of_eqI: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "set (map fst xs) = set (map fst ys)"   and "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k" shows "map_of xs = map_of ys"
that_3: fixes x :: "'a"   and y :: "'b"   and z :: "'c" assumes "P x y \<and> Q y z" shows "R x z"
complemented_filter_inf_principal: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and top :: "'c"   and F :: "'c set"   and G :: "'c set"   and x :: "'c" assumes "class.distrib_lattice_top inf less_eq less sup top"   and "ord.filter less_eq F \<and> ord.filter less_eq G \<and> semilattice_inf.filter_sup inf less_eq F G = UNIV \<and> F \<inter> G = {top}" shows "\<exists>xa. F \<inter> {y. less_eq x y} = {y. less_eq xa y}"
add_neg_numeral_special_9: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) (uminus one) = uminus (numeral.numeral one plus (num.Bit0 num.One))"
domain_atlas_subset_carrier: fixes c :: "('a, 'c) chart" assumes "c \<in> src.atlas" shows "domain c \<subseteq> src.carrier"
WT_gpv_bindD2: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv"   and x :: "'g" assumes "\<Gamma> \<turnstile>g gpv \<bind> f \<surd>"   and "x \<in> results_gpv \<Gamma> gpv" shows "\<Gamma> \<turnstile>g f x \<surd>"
adds_pp_iff: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "(s adds t) = (pp.mapping_of s adds pp.mapping_of t)"
nn_integral_count_space_finite: fixes A :: "'d set"   and f :: "'d \<Rightarrow> ennreal" assumes "finite A" shows "integral\<^sup>N (count_space A) f = sum f A"
bigo_const4: fixes c :: "'c" assumes "c \<noteq> (0::'c)" shows "O(\<lambda>x. 1::'c) \<subseteq> O(\<lambda>x. c)"
finite_update_induct: fixes f :: "'b \<Rightarrow> 'c"   and c :: "'c"   and P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool" assumes "finite {a. f a \<noteq> c}"   and "P (\<lambda>a. c)"   and "\<And>a b f. \<lbrakk>finite {a. f a \<noteq> c}; f a = c; b \<noteq> c; P f\<rbrakk> \<Longrightarrow> P (f(a := b))" shows "P f"
check'_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and dir :: "('a, 'b) Direction"   and x\<^sub>i :: "nat"   and s :: "('a, 'b) state" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update" shows "PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update dir x\<^sub>i s \<equiv> let l\<^sub>i = the (Direction.LB dir s x\<^sub>i); x\<^sub>j' = min_rvar_incdec_eq dir s (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i) in case x\<^sub>j' of Inl I \<Rightarrow> set_unsat I s | Inr x\<^sub>j \<Rightarrow> pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s"
assms: shows "\<forall>x xi. P x \<and> hfsynth_ID_R (fst R) x \<longrightarrow> hn_refine (emp * hn_ctxt (fst R) x xi) (f $ xi) (emp * hn_ctxt (snd R) x xi) S (g $ x)"
rr_def: shows "rr = real_roots_of_int_poly p"
1: shows "finite ({v. llength v = enat n} \<times> UNIV)"
invertible_mat_n: fixes n :: "'d" assumes "n \<noteq> (0::'d)" shows "invertible (mat n)"
simps_3: fixes var :: "'f"   and t :: "('d, 'f) trm" shows "SIGP (DiffAssign var t) = {Inl x |x. x \<in> SIGT t}"
DERIV_inverse': fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(f has_field_derivative D) (at x within s)"   and "f x \<noteq> (0::'a)" shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
n_def2: shows "d (n + 1) powr 2 ^ (n + 1) * (\<Prod>j = 1..n. real_of_int (a j)) * (1 / as) \<le> real_of_int (a (n + 1))"
boundOuputLengthSimp2: fixes \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi" shows "residualLength (\<alpha> \<prec> P) = length (bn \<alpha>)"
box_stp_act: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> (\<box>$P) = (\<box>P)"
finite_refines_finite: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "finite (A // R)"   and "R \<subseteq> S"   and "equiv A R"   and "equiv A S" shows "finite (A // S)"
pmf_eq_iff: fixes M :: "'d pmf"   and N :: "'d pmf" shows "(M = N) = (\<forall>i. pmf M i = pmf N i)"
utp_order_def: fixes H :: "'a hrel \<Rightarrow> 'a hrel" shows "utp_order H = \<lparr>carrier = \<lbrakk>H\<rbrakk>\<^sub>H, eq = (=), le = (\<sqsubseteq>)\<rparr>"
valid_index_weave_3: fixes is1 :: "nat list"   and ds :: "nat list"   and A :: "nat set"   and is2 :: "nat list" assumes "is1 \<lhd> nths ds A"   and "is2 \<lhd> nths ds (- A)" shows "nths (weave A is1 is2) (- A) = is2"
simps: fixes a1 :: "32 Word.word"   and a2 :: "real" shows "(a1 \<equiv>\<^sub>E a2) = ((\<exists>r. a1 = POS_INF \<and> a2 = r \<and> real_of_int (sint POS_INF) \<le> r) \<or> (\<exists>r. a1 = NEG_INF \<and> a2 = r \<and> r \<le> real_of_int (sint NEG_INF)) \<or> (\<exists>w. a1 = w \<and> a2 = real_of_int (sint w) \<and> real_of_int (sint w) < real_of_int (sint POS_INF) \<and> real_of_int (sint NEG_INF) < real_of_int (sint w)))"
LN_7: fixes arb_labs :: "t list list"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es : ts _> ts'" shows "\<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es''' : ts _> ts'"
two_comparisons_into_case_order_5: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and Q :: "'c"   and P :: "'c"   and R :: "'c" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if le_of_comp comp x y then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
lookup_some_in_dom: fixes \<rho> :: "(nat \<times> exp) list"   and x :: "nat"   and v :: "exp" assumes "lookup \<rho> x = Some v"   and "v \<noteq> EVar x" shows "x \<in> sdom \<rho>"
seg_sym: fixes a :: "'a"   and b :: "'a" shows "segment a b = segment b a"
snoc_def: fixes n :: "'a"   and bs :: "bool list"   and i :: "nat"   and P :: "'a fset" shows "snoc n bs i P = (if bs ! i then finsert n P else P)"
mhead_def: fixes m :: "methd" shows "mhead m = \<lparr>access = access m, static = static m, pars = pars m, resT = resT m\<rparr>"
foldl_length_aux: fixes a :: "nat"   and l :: "'a list" shows "foldl (\<lambda>i x. Suc i) a l = a + length l"
coretract_property: fixes y :: "'b" assumes "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "y \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y)"
verit_sum_simplify: fixes a :: "'d" shows "a + (0::'d) = a"
verit_minus_simplify_4: fixes b :: "'a" shows "- (- b) = b"
precond_equiv: fixes P :: "('b, 'c) urel" shows "(P = P ;; true) = (out\<alpha> \<sharp> P)"
bot_eq_sup_iff: fixes x :: "'c multiset"   and y :: "'c multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
abs_eq: fixes x :: "'b" shows "prod_left x = CBlinfun (\<lambda>a. a ** x)"
hyp_edge_from_def: fixes c :: "'form"   and "is" :: "nat list"   and n :: "nat"   and s :: "'subst" shows "hyp_edge_from c is n s = ((c, 0 # hyp_port_path_for (it' c) is (subst s (freshen n anyP))), hyp_port_h_for (it' c) is (subst s (freshen n anyP)))"
typing_inversion_8: fixes \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes "\<Gamma> \<turnstile> FALSE : T" obtains "T = tBOOL"
d1: shows "(r|labels \<tau> @ labels \<sigma>'|, r|labels \<tau>| + r|labels \<sigma>|) \<in> mul_eq r"
has_domain_iff_arr: fixes f :: "'a arrow_of_spans_data" shows "(V.domains f \<noteq> {}) = V.arr f"
substAbs_psubstAbs_compose: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "goodEnv rho" shows "A $[rho] $[Y / y]_ys = A $[rho &[Y / y]_ys]"
Abs_assn_inverse: fixes y :: "heap \<times> nat set \<Rightarrow> bool" assumes "y \<in> Collect proper" shows "Rep_assn (Abs_assn y) = y"
main: shows "w \<in> {1..2 ^ (LENGTH(32) - 1) - 1}"
Rep_process_inverse: fixes x :: "'a process" shows "Abs_process (Rep_process x) = x"
sel: fixes x2 :: "'a" shows "the (Some x2) = x2"
mult_scalar_monom_mult: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.mult_scalar (MPoly_Type_Class.punit.monom_mult c t p) q = MPoly_Type_Class.punit.monom_mult c t (MPoly_Type_Class.punit.mult_scalar p q)"
eta_expansion_safe': fixes n :: "nat"   and S :: "stack_elem list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" assumes "n \<le> arg_prefix S" shows "(\<Gamma>, eta_expand n e, S) \<Rightarrow>\<^sup>* (\<Gamma>, e, S)"
leg0_simps_3: shows "trg\<^sub>D (F f) = src\<^sub>D (F r)"
compact_space_subtopology: fixes X :: "'a topology"   and S :: "'a set" assumes "compactin X S" shows "compact_space (subtopology X S)"
f0: fixes i :: "nat" assumes "i \<noteq> 0" shows "(f i = 0) = (k < i)"
split_sel_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = Some (the option) \<and> \<not> P (f2 (the option))))"