hcju/isabelleps huggingface dataset:
Nil_in_Derivs_iff: fixes w :: "'a list"   and A :: "'a list set" shows "([] \<in> Derivs w A) = (w \<in> A)"
dims0_def: shows "st.dims0 = nths dims st.vars0"
G_face_cycle_succ_a: shows "G.face_cycle_succ a = a"
LTL_suffix_G: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and i :: "nat" assumes "w \<Turnstile> G \<phi>" shows "suffix i w \<Turnstile> G \<phi>"
mult_ket_vec_is_ket_vec_of_mult: fixes A :: "complex mat"   and v :: "complex vec" assumes "dim_col A = dim_vec v" shows "|col_fst (A * |v\<rangle>)\<rangle> = A * |v\<rangle>"
abs_norm_cancel: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and a :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a\<bar> = norm a"
star_induct: fixes r :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and x1a :: "'a"   and x1b :: "'b"   and x2a :: "'a"   and x2b :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "star r (x1a, x1b) (x2a, x2b)"   and "\<And>a b. P a b a b"   and "\<And>a b aa ba ab bb. \<lbrakk>r (a, b) (aa, ba); star r (aa, ba) (ab, bb); P aa ba ab bb\<rbrakk> \<Longrightarrow> P a b ab bb" shows "P x1a x1b x2a x2b"
negligible_eq_zero_density_alt: fixes S :: "'b set" shows "negligible S = (\<forall>x\<in>S. \<forall>e>0. \<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue U < e * Sigma_Algebra.measure lebesgue (ball x d))"
prefix_comparable_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<bowtie> v \<equiv> u \<le>p v \<or> v \<le>p u"
select_weight_drop_zero: fixes xs :: "(natural \<times> 'a) list" shows "Random.select_weight (filter (\<lambda>(k, uu_). 0 < k) xs) = Random.select_weight xs"
app_pair_spp_of_vec_of: fixes p :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat" assumes "spp_inv_pair p" shows "app_pair spp_of (app_pair vec_of p) = p"
sep_conj_impl: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a"   and P' :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "(P \<and>* Q) h"   and "\<And>h. P h \<Longrightarrow> P' h"   and "\<And>h. Q h \<Longrightarrow> Q' h" shows "(P' \<and>* Q') h"
strict_coboundedI2: fixes c :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "c \<subset># b" shows "c \<subset># a \<union># b"
rel_map_1: fixes Sb :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and i :: "'j \<Rightarrow> 'h"   and x :: "'j pmf"   and y :: "'i pmf" shows "rel_pmf Sb (map_pmf i x) y = rel_pmf (\<lambda>x. Sb (i x)) x y"
inconsist_init_converg_less: fixes f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat"   and m :: "nat" assumes "f \<in> \<R>"   and "\<phi> i \<in> \<R>"   and "inconsist i (f \<triangleright> n) \<down>"   and "the (inconsist i (f \<triangleright> n)) < Suc n" shows "inconsist i (f \<triangleright> (n + m)) = inconsist i (f \<triangleright> n)"
emeasure_pair_measure_finite: fixes A :: "('a \<times> 'b) set" assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"   and "emeasure (M1 \<Otimes>\<^sub>M M2) A < \<infinity>" shows "AE x in M1. emeasure M2 {y \<in> space M2. (x, y) \<in> A} < \<infinity>"
lt_slength_0_2: fixes xs :: "[:'r:]" assumes "xs \<noteq> \<bottom>" shows "lt\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>xs + 1) = TT"
weakly_future_consistent_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "weakly_future_consistent P I D R \<equiv> \<forall>u\<in>range D \<inter> (- I) `` range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> next_dom_events P D u xs = next_dom_events P D u ys \<and> ref_dom_events P D u xs = ref_dom_events P D u ys"
induced_morphism_eq: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e"   and x :: "'d"   and C :: "('f set \<times> 'g) set" assumes "ind_mor_btw_stalks X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f x"   and "C \<in> stalk.carrier_stalk is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y (f x)" obtains V :: "'f set" and q :: "'g" where "(V, q) \<in> C" and "\<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub> C = direct_lim.class_of \<O>\<^sub>X \<rho>\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} (f \<^sup>\<inverse> X V) (\<phi>\<^sub>f V q)"
0: shows "substT (Var y) (Var u) x = Var y"
analz_Un_implInsecSet: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({sym.implInsec A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
simps: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a stream" shows "PTA_Reachability.pairwise P a = (\<exists>aa b xs. a = aa ## b ## xs \<and> P aa b \<and> PTA_Reachability.pairwise P (b ## xs))"
primitive_principal_form: assumes "\<chi> \<noteq> principal_dchar n" shows "\<exists>\<Phi>. primitive_dchar conductor \<Phi> \<and> (\<forall>n. \<chi> n = \<Phi> n * principal_dchar n n)"
pf_1: shows "Ball (set (prime_factorization_nat n)) prime"
zero_in_bigo: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "(\<lambda>_. 0::'c) \<in> O[F](f)"
holomorphic_injective_imp_regular: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "inj_on f S"   and "\<xi> \<in> S" shows "deriv f \<xi> \<noteq> 0"
continuous_on_subset: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and t :: "'d set" assumes "continuous_on s f"   and "t \<subseteq> s" shows "continuous_on t f"
length_Tensor: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and M1 :: "'b list list"   and M2 :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "length (mult.Tensor f M1 M2) = length M1 * length M2"
set_plus_mono4_b: fixes a :: "'c"   and C :: "'c set"   and x :: "'c"   and D :: "'c set" assumes "a \<in> C"   and "x \<in> a +o D" shows "x \<in> D + C"
sup_state_conv: fixes P :: "'c prog"   and ST :: "ty list"   and LT :: "ty err list"   and ST' :: "ty list"   and LT' :: "ty err list" shows "P \<turnstile> (ST, LT) \<le>\<^sub>i (ST', LT') = (P \<turnstile> ST [\<le>] ST' \<and> P \<turnstile> LT [\<le>\<^sub>\<top>] LT')"
blinfun_apply_inverse: fixes x :: "'a \<Rightarrow>\<^sub>L 'b" shows "Blinfun (blinfun_apply x) = x"
of_nat_le_0_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
simps_95: fixes x61 :: "'a formula"   and x62 :: "'a formula" shows "atoms (x61 \<^bold>\<rightarrow> x62) = atoms x61 \<union> atoms x62"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and var :: "'a"   and val :: "bool" shows "restrict_top (IF v t e) var val = (if v = var then if val then t else e else IF v t e)"
FormP_Eq: fixes x :: "tm"   and y :: "tm" shows "{TermP x, TermP y} \<turnstile> FormP (Q_Eq x y)"
mono_monitor: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<pi>' :: "(String.literal \<times> event_data list) prefix" assumes "\<pi> \<le> \<pi>'" shows "M \<pi> \<subseteq> M \<pi>'"
TUA: shows "FUadmitFO \<sigma> \<phi> U"
finetely_gen: fixes I :: "'a set" assumes "ideal I R" shows "\<exists>A\<subseteq>carrier R. finite A \<and> I = Idl A"
1: shows "Cring_Poly.compose R (T\<^bsub>a\<^esub> f) (X_poly_minus R a) = monom P (T\<^bsub>a\<^esub> f 0) 0 \<oplus>\<^bsub>P\<^esub> X_poly_minus R a \<otimes>\<^bsub>P\<^esub> Cring_Poly.compose R (poly_shift (T\<^bsub>a\<^esub> f)) (X_poly_minus R a)"
is_element_ptr_kind_none: fixes node_ptr :: "(_) node_ptr" assumes "\<not> is_element_ptr_kind node_ptr" shows "cast node_ptr = None"
prv_neg_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "bprv \<phi>"   and "bprv (neg \<phi>)" shows "bprv fls"
inversion_number_append: fixes xs :: "'a list"   and ys :: "'a list" shows "inversion_number (xs @ ys) = inversion_number xs + inversion_number ys + inversion_number_between xs ys"
iIN_div_if: fixes k :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "[n\<dots>,d] \<oslash> k = [n div k\<dots>,d div k + (if n mod k + d mod k < k then 0 else Suc 0)]"
VV_cases: fixes v :: "'a"   and p :: "Player" assumes "v \<in> V" obtains "v \<in> VV p"   | "v \<in> VV p**"
arr_RelD_4: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "vcard T = 3\<^sub>\<nat>"
abs_benv_upd: fixes \<beta> :: "nat \<Rightarrow> contour option"   and c :: "nat"   and b :: "contour" shows "|\<beta>(c \<mapsto> b)| = |\<beta>|(c \<mapsto> |b|)"
induct: fixes P :: "num \<Rightarrow> num \<Rightarrow> bool"   and a0 :: "num"   and a1 :: "num" assumes "P num.One num.One"   and "\<And>m. P num.One (num.Bit0 m)"   and "\<And>m. P num.One (num.Bit1 m)"   and "\<And>n. P (num.Bit0 n) num.One"   and "\<And>n m. P n m \<Longrightarrow> P (num.Bit0 n) (num.Bit0 m)"   and "\<And>n m. P n m \<Longrightarrow> P (num.Bit0 n) (num.Bit1 m)"   and "\<And>n. P (num.Bit1 n) num.One"   and "\<And>n m. P n m \<Longrightarrow> P (num.Bit1 n) (num.Bit0 m)"   and "\<And>n m. P n m \<Longrightarrow> P (num.Bit1 n) (num.Bit1 m)" shows "P a0 a1"
disjnt_Sigma_iff: fixes A :: "'c set"   and C :: "'c \<Rightarrow> 'd set"   and B :: "'c set" shows "disjnt (Sigma A C) (Sigma B C) = ((\<forall>i\<in>A \<inter> B. C i = {}) \<or> disjnt A B)"
assms_4: shows "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
axioms_2: fixes G :: "('b, 'c) monoid_scheme" assumes "comm_group G" shows "Group.group G"
UP_domain_def: fixes R :: "('a, 'b) ring_scheme" shows "UP_domain R \<equiv> domain R"
vector_space_row_space: fixes A :: "rat mat"   and nr :: "nat" assumes "A \<in> carrier_mat nr n" shows "vectorspace class_ring (gs.vs (gs.row_space A))"
L_transform_formula_eqvt: fixes p :: "perm"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "p \<bullet> L_transform_formula x = L_transform_formula (p \<bullet> x)"
blinding_of_on_blindable: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "blinding_of_on A h bo" shows "blinding_of_on {x. set1_blindable\<^sub>m x \<subseteq> A} (hash_blindable h) (blinding_of_blindable h bo)"
all_not_in_conv: fixes A :: "'g set" shows "(\<forall>x. x \<notin> A) = (A = {})"
xc: fixes m :: "nat" assumes "0 < m" shows "real m * x \<le> c"
f_def: shows "f \<equiv> SOME f. u = #v * #w + Polynomial.smult (of_nat q) (#f)"
linear_inverse_left: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and f' :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "linear (*s) (*s) f"   and "linear (*s) (*s) f'" shows "(f \<circ> f' = id) = (f' \<circ> f = id)"
simps_2: fixes p :: "perm"   and t :: "('a, 'b, 'c, 'd) Tree" shows "p \<bullet> tNot t = tNot (p \<bullet> t)"
RID\<^sub>G_def: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" shows "RID\<^sub>G s \<equiv> dom s \<union> \<Union> (RID\<^sub>L ` ran s)"
eq: shows "local.punit.monom_mult (lc q') (0::'a) (rep_list p') = local.punit.monom_mult (lc p') (0::'a) (rep_list q')"
graph_path_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list" shows "graph_path \<alpha>e \<alpha>n invar inEdges' \<equiv> graph (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar \<and> graph_nodes_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g. foldri (\<alpha>n g)) \<and> graph_pred_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g n. foldri (inEdges' g n))"
cycle_Nil: fixes G :: "('a, 'b) graph_scheme" shows "cycle G [] = False"
HComplex_diff: fixes x1 :: "real star"   and y1 :: "real star"   and x2 :: "real star"   and y2 :: "real star" shows "HComplex x1 y1 - HComplex x2 y2 = HComplex (x1 - x2) (y1 - y2)"
i: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "\<phi> \<pi>l"   and "until \<phi> \<psi> (map stl \<pi>l)"   and "\<chi> (map stl \<pi>l)" shows "\<chi> \<pi>l"
c0: shows "lookup p (t \<oplus> lt q) / lc q \<noteq> (0::'b)"
possible_mono: fixes t :: "'b ttree"   and t' :: "'b ttree"   and x :: "'b" assumes "paths t \<subseteq> paths t'"   and "possible t x" shows "possible t' x"
L4_2b: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" assumes "ipath gba.E \<sigma>"   and "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)"   and "gba.is_acc \<sigma>" shows "\<exists>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i)) \<and> \<eta> \<in> old (\<sigma> j)"
pr1_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.comp (pr1 (cod f) (cod g)) (tuple f g) = f"
THE_addr_loc_type: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and T :: "ty" assumes "P,h \<turnstile> a@al : T" shows "(THE T. P,h \<turnstile> a@al : T) = T"
less_asym: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y"   and "\<not> P \<Longrightarrow> less y x" shows "P"
PLemmas_3: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set"   and p :: "int" shows "C (AllowPortFromTo x y p) = allow_from_to_port p x y"
prepend_ltake_ldrop_id: fixes i :: "nat"   and t :: "'d llist" shows "prepend (Co_Snapshot.ltake i t) (Co_Snapshot.ldrop i t) = t"
Abs_ell2_inverse: fixes y :: "'c \<Rightarrow> complex" assumes "y \<in> {x. has_ell2_norm x}" shows "Rep_ell2 (Abs_ell2 y) = y"
intro: fixes letter_eq :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and alphabet1 :: "'b list"   and alphabet2 :: "'c list" assumes "\<And>a b. letter_eq a b \<Longrightarrow> (a \<in> set alphabet1) = (b \<in> set alphabet2)" shows "DFAs_axioms alphabet1 alphabet2 letter_eq"
simps_2: fixes t :: "'b tree" shows "left (mirror t) = mirror (right t)"
Id_on_Gr: fixes A :: "'b set" shows "Id_on A = BNF_Def.Gr A id"
supp_conv_fresh: fixes x :: "'a" shows "supp x = {a. \<not> a \<sharp> x}"
unit_factor_primepow: fixes p :: "'a" assumes "primepow p" shows "unit_factor p = (1::'a)"
ideal_resource_wiring: fixes callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'd, auth_query, nat option) gpv"   and s :: "'d" shows "CNV callee s |\<^sub>= 1\<^sub>C \<rhd> S_CHAN.res = RES (callee_sec_channel callee \<oplus>\<^sub>O \<dagger>\<dagger>S_CHAN.send_oracle \<oplus>\<^sub>O \<dagger>\<dagger>S_CHAN.recv_oracle) (s, (), Void)"
assms: shows "\<forall>(pc', \<tau>')\<in>set (norm_eff i P pc \<tau>). pc' < mpc"
ope: fixes U :: "'a set" assumes "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"
assms_1: assumes "nofail a"   and "nofail b" shows "a \<le>\<^sub>n m"
c_failuresp_c_failures_eq: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "c_failuresp step out s\<^sub>0 = (\<lambda>x xa. (x, xa) \<in> c_failures step out s\<^sub>0)"
A\<^sub>A_asm_mem: shows "\<forall>x. case A\<^sub>A_of A\<^sub>C x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>A_of mem\<^sub>1\<^sub>C x \<noteq> v \<or> mem\<^sub>A_of mem\<^sub>2\<^sub>C x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written (mds\<^sub>A_of mds\<^sub>C) x"
transfer: shows "rel_fun pcr_real pcr_real (\<lambda>X n. - X n) uminus"
inside_inside: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> inside T" shows "inside S - T \<subseteq> inside T"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'c" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> toy_init i | Some s \<Rightarrow> s, snd \<sigma>)"
pfa_shrink_cap_def: fixes k :: "nat" shows "pfa_shrink_cap k \<equiv> \<lambda>(a, n). array_shrink a k \<bind> (\<lambda>a'. return (a', min k n))"
complex_cos_eq: fixes w :: "complex"   and z :: "complex" shows "(cos w = cos z) = (\<exists>n\<in>\<int>. w = z + complex_of_real (2 * n * pi) \<or> w = - z + complex_of_real (2 * n * pi))"
h6: shows "get_matrix_R (reduce_system_R p (qs, matrix, subsets, signs)) = M_mat_R (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs))) (get_subsets_R (reduce_system_R p (qs, matrix, subsets, signs)))"
lprefix_llength_lprefix: fixes a :: "'b llist"   and c :: "'b llist"   and b :: "'b llist" assumes "lprefix a c"   and "lprefix b c"   and "llength a \<le> llength b" shows "lprefix a b"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "pheap (Hp x hs) = (\<forall>h\<in>set hs. Multiset.Ball (mset_heap h) ((\<le>) x) \<and> pheap h)"
simps_1: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (x[i] ::= a, s) = Some (SKIP, s(x := (s x)(aval i s := aval a s)))"
cSup_eq_maximum: fixes z :: "'c"   and X :: "'c set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" shows "Sup X = z"
1: shows "Dom \<nu>' = Cod \<nu> \<and> Dom \<mu>' = Cod \<mu>"
cod_simp: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "\<F>\<^sub>SC.cod f = \<F>C.cod f"
replication_numbers_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection" shows "incidence_system.replication_numbers point_set block_collection \<equiv> {block_collection rep x |x. x \<in> point_set}"
simps: fixes p :: "(action \<times> nat) list"   and n :: "nat" shows "shift p n = map (\<lambda>(a, s). (a, if s = 0 then 0 else s + n)) p"
assms: shows "u @ v\<^sub>1 =\<^sub>F u @ v\<^sub>2"
run_V0: fixes r :: "nat \<Rightarrow> 'a" assumes "a.is_run r" shows "r 0 \<in> a.V0"
rev_eq_Cons_iff: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
empty_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_set A {} {}"
simps_5: fixes x2 :: "jump"   and x1 :: "xcpt" shows "Jump x2 \<noteq> Xcpt x1"
collapse: fixes fundef :: "'b fundef" shows "Fundef (body fundef) (arity fundef) = fundef"
higher_differentiable_on_addD: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" assumes "higher_differentiable_on S f (n + m)" shows "higher_differentiable_on S f n"
map_le_empty: fixes g :: "'c \<Rightarrow> 'd option" shows "Map.empty \<subseteq>\<^sub>m g"
eventually_const_iff: fixes P :: "bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. P) = (P \<or> F = bot)"
step: fixes x :: "name"   and xs :: "name list"   and P :: "('e, 'f, 'g) psi" shows "\<lparr>\<nu>*(x # xs)\<rparr>P = \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>*xs\<rparr>P)"
simps_3: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a" shows "heap_base.typeof_h typeof_addr h Null = typeof Null"
encCond: shows "\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))"
horner_composition_hom: fixes cs :: "int list"   and xs :: "int list" shows "horner_composition (map of_int cs) (map of_int xs) = of_int_poly (horner_composition cs xs)"
select_convs_4: fixes obj_mark :: "bool"   and obj_fields :: "'b \<Rightarrow> 'd option"   and obj_payload :: "'b \<Rightarrow> 'c option"   and more :: "'a" shows "object.more \<lparr>obj_mark = obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr> = more"
induct: fixes P :: "(nat \<times> nat list \<times> form) list \<Rightarrow> form \<Rightarrow> bool"   and a0 :: "(nat \<times> nat list \<times> form) list"   and a1 :: "form" assumes "\<And>s v xs. \<lbrakk>\<And>x. \<lbrakk>map_of s v = None; x \<in> set xs\<rbrakk> \<Longrightarrow> P s x; \<And>x2 x y xa. \<lbrakk>map_of s v = Some x2; (x, y) = x2; length x = length xs; xa \<in> set xs\<rbrakk> \<Longrightarrow> P s xa; \<And>x2 x y xa. \<lbrakk>map_of s v = Some x2; (x, y) = x2; length x \<noteq> length xs; xa \<in> set xs\<rbrakk> \<Longrightarrow> P s xa\<rbrakk> \<Longrightarrow> P s (Var v xs)"   and "\<And>s n. P s (LC n)"   and "\<And>s n xs. (\<And>x. x \<in> set xs \<Longrightarrow> P s x) \<Longrightarrow> P s (Op n xs)"   and "\<And>s n v f. \<lbrakk>\<And>x. \<lbrakk>v \<in> fv_subst s; x = fresh_for (fv_subst s)\<rbrakk> \<Longrightarrow> P ((v, [], Var x []) # s) f; v \<notin> fv_subst s \<Longrightarrow> P s f\<rbrakk> \<Longrightarrow> P s (Quant n v f)" shows "P a0 a1"
h2: fixes j :: "nat" assumes "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))" shows "j < length signs"
case_rews_1: fixes f :: "'e \<rightarrow> 'f \<rightarrow> 'g \<rightarrow> 'd" shows "Tuple3_case\<cdot>f\<cdot>\<bottom> = \<bottom>"
aff_dim_sums_Int: fixes S :: "'a set"   and T :: "'a set" assumes "affine S"   and "affine T"   and "S \<inter> T \<noteq> {}" shows "aff_dim {x + y |x y. x \<in> S \<and> y \<in> T} = aff_dim S + aff_dim T - aff_dim (S \<inter> T)"
is_digit_simps_1: fixes n :: "nat" assumes "n < 10" shows "is_digit n"
le_take_imp_prefix: fixes a :: "nat"   and b :: "nat"   and xs :: "'b list" assumes "a \<le> b" shows "prefix (xs \<down> a) (xs \<down> b)"
twice_freq_le_imp_minima: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat"   and u :: "'a tree"   and b :: "'a"   and a :: "'a" assumes "\<forall>c\<in>alphabet t. w\<^sub>a \<le> freq t c \<and> w\<^sub>b \<le> freq t c"   and "alphabet u = alphabet t \<union> {b}"   and "a \<in> alphabet u"   and "a \<noteq> b"   and "freq u = (\<lambda>c. if c = a then w\<^sub>a else if c = b then w\<^sub>b else freq t c)" shows "minima u a b"
mkbot_def: fixes L :: "(alt \<times> alt) set"   and b :: "alt" shows "mkbot L b \<equiv> {(x, y). x <\<^bsub>L\<^esub> y \<and> x \<noteq> b \<and> y \<noteq> b} \<union> {(b, y) |y. y \<noteq> b}"