hcju/isabelleps huggingface dataset:
csol_mem_csols: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "(ll_on_open.csol T f X t0 x0 t, t) \<in> ll_on_open.csols T f X t0 x0"
length_0_conv: fixes xs :: "'e list" shows "(length xs = 0) = (xs = [])"
deterministic: fixes x :: "'state"   and y :: "'state"   and z :: "'state" assumes "x \<rightarrow> y"   and "x \<rightarrow> z" shows "y = z"
DOWN_def: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "DOWN t \<equiv> the_elem (Dom t)"
xcpt_app_def: fixes i :: "'c instr"   and P :: "'d prog"   and pc :: "nat"   and mxs :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and \<tau> :: "ty list \<times> ty err list" shows "xcpt_app i P pc mxs xt \<tau> \<equiv> \<forall>(f, t, C, h, d)\<in>set (relevant_entries P i pc xt). (case C of None \<Rightarrow> True | \<lfloor>C'\<rfloor> \<Rightarrow> is_class P C') \<and> d \<le> length (fst \<tau>) \<and> d < mxs"
bal_d_delete: fixes t :: "'a tree"   and dl :: "nat"   and x :: "'a"   and t' :: "'a tree"   and dl' :: "nat" assumes "bal_d (size t) dl"   and "local.delete x (t, dl) = (t', dl')" shows "bal_d (size t') dl'"
apps_preserves_betas: fixes rs :: "lam list"   and ss :: "lam list"   and r :: "lam" assumes "rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss" shows "r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> r \<degree>\<degree> ss"
simps: fixes X :: "'a set" shows "Autoref_Misc.the_RES (RES X) = X"
preserves_ide: fixes a :: "'a" assumes "C.ide a" shows "C.ide (C.L a)"
CE'FMT: shows "ce' = csp + add_mset s cc"
sd_efficient_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "sd_efficient_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> sd_efficient_sds_axioms agents alts sds"
dom_complete_auxi: fixes i :: "nat"   and ss' :: "nat list"   and k :: "nat" assumes "i < length start"   and "dom_kildall start ! i = ss' \<and> k \<notin> set ss'" shows "non_strict_dominate k i"
T_simp: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "T (f, g) = f \<otimes> g"
assms_2: fixes w :: "'a stream"   and a :: "'b" assumes "R w a" shows "P a \<and> (\<exists>u v. w = u @- v \<and> u \<noteq> [] \<and> R v (fold f u a))"
exists_indexed_eval_monomial': fixes P :: "'e multiset \<Rightarrow> 'a"   and Qs :: "('e multiset \<Rightarrow> 'a) list"   and i :: "'e" assumes "carrier_coeff P"   and "list_all carrier_coeff Qs"   and "P \<noteq> indexed_const \<zero>"   and "list_all (\<lambda>Q. index_free Q i) Qs" obtains m :: "'e multiset" where "length Qs \<le> count m i" and "indexed_eval (P # Qs) i m \<noteq> \<zero>"
wt_Done: fixes s :: "'state"   and i :: "nat" shows "wt Done s i = 1"
WHILEI_def: shows "WHILEI \<equiv> iWHILEI (\<bind>) RETURN"
map_add_comm: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" assumes "dom m1 \<inter> dom m2 = {}" shows "m1 ++ m2 = m2 ++ m1"
proj_tuple_in_join_mask_idle: fixes n :: "nat"   and X :: "nat set"   and as :: "'a option list"   and pos :: "bool"   and t :: "'a option list set" assumes "wf_tuple n X as" shows "proj_tuple_in_join pos (join_mask n X) as t = join_cond pos t as"
powr_has_integral_at_top: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "0 < a + b"   and "c < - 1" shows "((\<lambda>x. (b + x) powr c) has_integral - ((b + a) powr (c + 1) / (c + 1))) {a<..}"
simps_2: fixes n :: "nat"   and s :: "'d stream" shows "stake (Suc n) s = shd s # stake n (stl s)"
pt_bij1: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "pi \<bullet> x = y" shows "x = rev pi \<bullet> y"
flip_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "flip P \<equiv> \<lambda>s s'. P s' s"
comp_maps: fixes \<eta> :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and \<theta> :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "Set_Theory.map \<eta> A B"   and "Set_Theory.map \<theta> B C" shows "Set_Theory.map (\<theta> \<circ> \<eta> \<down> A) A C"
prodag_elemfun: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and f :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. aGroup (A k)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A)" shows "f \<in> extensional I"
exists_one_iff_filter_one: fixes l :: "'d list"   and a :: "'d" shows "(\<exists>!j. j < length l \<and> l ! j = a) = (length (filter ((=) a) l) = 1)"
scale_left_diff_distrib: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and x :: "'c" assumes "module scale" shows "scale (a - b) x = scale a x - scale b x"
map'_2: fixes x :: "String.literal"   and l :: "tm list"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and v :: "'a" assumes "\<not> occurs_in_list x l" shows "semantics_list e f l = semantics_list (e(x := v)) f l"
L_eqvt': fixes p :: "perm"   and \<alpha> :: "'act"   and P :: "'effect fs_set"   and f :: "'effect" shows "p \<bullet> L (\<alpha>, P, f) = L (p \<bullet> \<alpha>, p \<bullet> P, p \<bullet> f)"
dom_def_aux: fixes x :: "'a" shows "d x = 1' \<cdot> x ; 1"
ipresIGWlsAll_def: fixes h :: "'x \<Rightarrow> 'y"   and hA :: "'z \<Rightarrow> 'aa"   and MOD :: "('ab, 'ac, 'ad, 'ae, 'af, 'ag, 'x, 'z, 'ah) model_scheme"   and MOD' :: "('ai, 'aj, 'ad, 'ae, 'ak, 'al, 'y, 'aa, 'am) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
inverse_less_iff_less_neg: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "b < (0::'a)" shows "(inverse a < inverse b) = (b < a)"
stretch_Galois: fixes m :: "'a \<Rightarrow> real"   and y :: "(real, 'a) vec"   and x :: "(real, 'a) vec" assumes "\<And>k. m k \<noteq> 0" shows "(y = (\<chi>k. m k * x $ k)) = ((\<chi>k. y $ k / m k) = x)"
flipped_reflections_in_lconjseq: fixes S :: "'b set"   and ss :: "'b list" assumes "CoxeterSystem S"   and "ss \<in> lists S" shows "order.greater_eq (set (lconjseq ss)) (CoxeterSystem.flipped_reflections S (sum_list ss))"
Im_complex_of_real: fixes z :: "real" shows "is_real (cor z)"
strip_while_dropWhile_commute: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "strip_while P (dropWhile Q xs) = dropWhile Q (strip_while P xs)"
distinct_69: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com"   and x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com" shows "Cond x51 x52 x53 \<noteq> While x61 x62"
S_spawn: fixes u :: "'a"   and q :: "'b"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and M :: "'d set"   and P :: "'b multiset"   and k :: "nat"   and P' :: "'b multiset" assumes "(u, Spawn q, v) \<in> edges fg"   and "(u, M, P) \<in> S_cs fg k"   and "P' \<subseteq># {#q#} + P"   and "size P' \<le> k" shows "(v, M, P') \<in> S_cs fg k"
paths_le_split_via_eq: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "paths_le (Suc k) P = paths_le k P \<union> paths_eq (Suc k) P"
concurrent_ops_commute_rearrange: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and interp :: "'c \<Rightarrow> 'd \<Rightarrow> 'd option"   and xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "happens_before hb_weak hb" shows "happens_before.concurrent_ops_commute hb interp (xs @ x # ys) = happens_before.concurrent_ops_commute hb interp (xs @ ys @ [x])"
eval_Mn_total: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list" assumes "recfn (length xs) (Mn n f)"   and "Partial_Recursive.total f" shows "eval (Mn n f) xs = (if \<exists>z. eval f (z # xs) \<down>= 0 then Some (LEAST z. eval f (z # xs) \<down>= 0) else None)"
AssLVar: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and A :: "assigned"   and vn :: "lname" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "nrm A = nrm E \<union> {vn}"   and "brk A = brk E" shows "Env\<turnstile> B \<guillemotright>\<langle>LVar vn:=e\<rangle>\<guillemotright> A"
b_def: shows "b = f \<turnstile> folding_g.\<C>"
upright_alt: fixes a :: "'c zmultiset" shows "upright a = (\<forall>t. 0 < zcount a t \<longrightarrow> supported_strong a t)"
add_col_sub_row_carrier_1: fixes a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_row (add_col_sub_row a k l A) = dim_row A"
reachable_nonterminating: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and y :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "reachable x y"   and "nonterminating y" shows "nonterminating x"
InitConf_HAInitState_HARoot: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> InitConf A"   and "S \<noteq> HAInitState A" shows "S \<notin> States (HARoot A)"
measurable_product_then_coordinatewise: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'c measure"   and i :: "'d" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f x i) \<in> borel_measurable M"
infer_simp_2: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a" shows "infer \<Gamma> (Var x) = \<Gamma> x"
less_neg_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus a) = less a zero"
card_eq_SucD: fixes A :: "'c set"   and k :: "nat" assumes "card A = Suc k" shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
EphK_Aenc: fixes X :: "ephkey"   and X' :: "msg"   and Y' :: "msg" shows "EphK X \<noteq> Aenc X' Y'"
representsI: fixes r :: "'b \<Rightarrow>\<^sub>0 'a"   and B :: "'b set"   and x :: "'b" assumes "keys r \<subseteq> B"   and "rep r = x" shows "represents B r x"
dup_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<d>[a] : a \<rightarrow> a \<otimes> a\<guillemotright>"
Card_csucc: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "Card (csucc \<kappa>)"
bij_betw_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
trm_soundness: fixes rhs :: "'b trm set"   and X :: "'b list set" assumes "finite rhs" shows "lang_rhs {Trn X r |r. Trn X r \<in> rhs} = X \<cdot> lang (\<Uplus>{r. Trn X r \<in> rhs})"
notn: fixes k :: "nat" assumes "k \<le> l" shows "\<pi> k \<noteq> n"
no_deadlock: fixes cfg :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b" assumes "\<not> final cfg" shows "\<exists>cfg'. step cfg cfg'"
distinct_5: fixes name1' :: "name"   and name2' :: "name"   and pi' :: "pi" shows "\<zero> \<noteq> name1'<name2'>.pi'"
min_galleryD_gallery: fixes xs :: "'a set list" assumes "domain.min_gallery xs" shows "domain.gallery xs"
wf_subst_subst_idem: fixes \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "subst_idem \<theta>"
insert_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" shows "(a \<in> {b} \<union> A) = (a = b \<or> a \<in> A)"
unitary_proj_psi_minus_1: shows "unitary (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N)"
first_t_fusion: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" assumes "Dioid_Models_New.last x = first y" shows "first (t_fusion x y) = first x"
\<alpha>_HG: shows "\<alpha> H \<le> \<alpha> G"
AddSA_Events: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "HAEvents (HA [++] (S, SA)) = HAEvents HA \<union> SAEvents SA"
arr_cod: fixes f :: "'a" assumes "arr f" shows "arr (cod f)"
lookup_pair_eqI: fixes xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs"   and "cmp2.oalist_inv_raw ys"   and "set xs = set ys" shows "lookup_pair xs = cmp2.lookup_pair ys"
ag_carrier_carrier: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "carrier (b_ag A) = carrier A"
assms: shows "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vconverse A"
join_def: fixes A :: "('b \<times> 'b) set" shows "A\<^sup>\<down> = A\<^sup>* O (A\<inverse>)\<^sup>*"
build_rec_isSubobj_rev: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and D :: "char list"   and Cs :: "char list list"   and C :: "char list" assumes "is_subobj P (D, D # rev Cs)"   and "P \<turnstile> C \<prec>\<^sub>R D" shows "is_subobj P (C, C # D # rev Cs)"
has_integral_integral: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" shows "(f integrable_on s) = (f has_integral integral s f) s"
crsp: shows "crsp ly (as, lm) (s, l, r) ires"
assms_2: shows "ab = [a, b]\<^sub>\<circ>"
exhaust: fixes y :: "taylor_model" obtains x1 :: "float poly" and x2 :: "float interval" where "y = TaylorModel x1 x2"
binaryset_def: fixes A :: "'b set"   and B :: "'b set" shows "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
d: shows "(c \<circ> \<sigma>) \<longlonglongrightarrow> d"
bs_empty_def: shows "BsSkewBinomialHeapStruc.bs_empty \<equiv> Inl ()"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<rho>\<sigma>.map a = B (\<sigma> a) (\<rho> a)"
Neg_D: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> Neg A"   and "H \<turnstile> A" shows "H \<turnstile> B"
monoidal_language_axioms: shows "monoidal_language (\<cdot>)"
isNSContCR_cmod: fixes a :: "complex" shows "isNSCont cmod a"
finiteI_independent: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and B :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "\<not> module.dependent scale B" shows "finite B"
add_closed: fixes a :: "'vs"   and b :: "'vs" assumes "a \<in> U\<^sub>V\<^sub>S"   and "b \<in> U\<^sub>V\<^sub>S" shows "a +\<^sub>V\<^sub>S b \<in> U\<^sub>V\<^sub>S"
ord_eq_less_trans: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a = b"   and "b \<prec>\<^sub>t c" shows "a \<prec>\<^sub>t c"
simps_2: fixes m_start :: "nat"   and m_end :: "nat"   and i_start :: "nat"   and i_end :: "nat"   and blocks :: "(nat \<times> nat) list" shows "??.Jordan_Normal_Form_Existence.find_largest_block (m_start, m_end) ((i_start, i_end) # blocks) = (if m_end - m_start \<le> i_end - i_start then ??.Jordan_Normal_Form_Existence.find_largest_block (i_start, i_end) blocks else ??.Jordan_Normal_Form_Existence.find_largest_block (m_start, m_end) blocks)"
MSB_greater: fixes i :: "nat"   and Is :: "nat fset list"   and p :: "nat" assumes "i < length Is"   and "p |\<in>| Is ! i" shows "p < MSB Is"
l3_inv3D: fixes x :: "l3_state" assumes "x \<in> l3_inv3" shows "analz (ik x) \<subseteq> synth (analz (ik x \<inter> payload \<union> ik x \<inter> valid \<union> ik x \<inter> range LtK \<union> Tags))"
poincare_between_sum_distances: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_between u v w = (poincare_distance u v + poincare_distance v w = poincare_distance u w)"
IH: fixes B :: "'a set" assumes "B \<subset> A" shows "map_pmf ipl (random_bst B) = rqs_cost (card B)"
eq_dffI: fixes x :: "'b fract"   and y :: "'b fract" assumes "divides_ff x y"   and "divides_ff y x" shows "x =dff y"
ex_safe_get_and: fixes \<phi> :: "formula" assumes "safe_formula \<phi>" shows "list_ex safe_formula (get_and_list \<phi>)"
elims: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and y :: "('a \<times> 'b) multiset" assumes "BinomialHeapStruc.tree_to_multiset x = y" obtains e :: "'a" and a :: "'b" and r :: "nat" and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = BinomialHeapStruc.Node e a r ts" and "y = {#(e, a)#} + BinomialHeapStruc.queue_to_multiset ts"
fdomain_small: fixes r :: "V" shows "small {a. \<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
intros_8: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Decision X\<rangle> \<Rightarrow> Decision X"
poly_eq_by_eval: fixes P :: "'b poly"   and Q :: "'b poly" assumes "\<And>x. poly P x = poly Q x" shows "P = Q"
ENF_offending_set_P_representation: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P"   and "F \<in> set_offending_flows G nP" shows "F = {(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}"
m_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "x \<oplus>\<^bsub>S\<^esub> y = y \<oplus>\<^bsub>S\<^esub> x"
nv_2: shows "normal_vector D \<noteq> 0\<^sub>v n"
element_exists: fixes A :: "'a set" assumes "A \<in> contained.Partition" shows "\<exists>a\<in>G. a \<in> A"
Un_absorb: fixes A :: "'c set" shows "A \<union> A = A"
sigma_finite_measure_restrict_space: fixes M :: "'b measure"   and A :: "'b set" assumes "sigma_finite_measure M"   and "A \<in> sets M" shows "sigma_finite_measure (restrict_space M A)"
sproduct_def: fixes s1 :: "'c stream"   and s2 :: "'d stream" shows "sproduct s1 s2 = smerge (smap (\<lambda>x. smap (Pair x) s2) s1)"
prems_5: fixes x :: "'node" assumes "x \<in> set ns" shows "v \<notin> oldDefs g x"
exhaust: fixes y :: "[:'a:]" obtains "y = \<bottom>"   | "y = [::]"   | a :: "'a" and slist :: "[:'a:]" where "y = a :# slist" and "a \<noteq> \<bottom>" and "slist \<noteq> \<bottom>"
residual_network_sel_3: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_network \<Delta>" shows "source (countable_network.residual_network \<Delta> f) = source \<Delta>"
bit_word_of_int_iff: fixes k :: "int"   and n :: "nat" shows "bit (word_of_int k) n = (n < LENGTH('b) \<and> bit k n)"
simps_2: fixes c :: "color"   and uu :: "color"   and l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "paint c (Branch uu l k v r) = Branch c l k v r"
when_uminus_distrib: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and P :: "bool" assumes "class.group_add minus plus zero uminus" shows "zero.when zero (uminus a) P = uminus (zero.when zero a P)"
bindET_def: shows "bindET = (\<Lambda> m k. ErrorT\<cdot>(runErrorT\<cdot>m \<bind> (\<Lambda> n. case n of Err\<cdot>e \<Rightarrow> return\<cdot>(Err\<cdot>e) | Ok\<cdot>x \<Rightarrow> runErrorT\<cdot>(k\<cdot>x))))"
eventually_mult_Gcd: fixes S :: "nat set"   and s :: "nat" assumes "\<And>s t. \<lbrakk>s \<in> S; t \<in> S\<rbrakk> \<Longrightarrow> s + t \<in> S"   and "s \<in> S"   and "0 < s" shows "\<forall>\<^sub>F m in sequentially. m * Gcd S \<in> S"
op_atLeastAtMost_appr_def: shows "op_atLeastAtMost_appr = atLeastAtMost"
q: shows "suffix t' t'' \<or> t' = ev # t''"
on_meridian_rep: fixes a :: "proj2" assumes "z_non_zero a"   and "proj2_incident a meridian" shows "\<exists>y. a = proj2_abs (vector [0, y, 1])"
max_value_of_not_const_less_bis: assumes "\<not> const 0"   and "\<not> const 1" shows "(cmod (of_int (\<Sum>k<2 ^ n. (- int 1) ^ (f k + 1))))\<^sup>2 < real (2 ^ (2 * n))"
FvarsT_Var: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and x :: "'a" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> var" shows "FvarsT (Var x) = {x}"
hassoc_map_rel_internal_def: fixes K :: "('e \<times> 'g) set"   and V :: "('f \<times> 'h) set" shows "hassoc_map_rel K V = hassoc_map_rel_raw O \<langle>K, V\<rangle>map_rel"
subst_type_perm_eq: fixes b :: "Syntax.tvar"   and t :: "ty"   and a :: "Syntax.tvar" assumes "atom b \<sharp> t" shows "subst_type t (Mu a t) a = subst_type ((a \<leftrightarrow> b) \<bullet> t) (Mu b ((a \<leftrightarrow> b) \<bullet> t)) b"
add_col_sub_row_def: fixes a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "add_col_sub_row a k l A = addrow (- a) k l (addcol a l k A)"
hyps_1: shows "set (bt_dfs val r) \<union> set |q| = set |bt_augment r q|"
fsmult_def: fixes a :: "'r"   and v :: "'v" shows "a \<sharp>\<cdot> v \<equiv> (a \<delta>\<delta> (0::'g)) \<cdot> v"
sup_Assertion_def: fixes p :: "'a Assertion"   and q :: "'a Assertion" shows "p \<squnion> q = abs_wpt ({\<cdot> p } \<squnion> {\<cdot> q })"
no_outstanding_write_to_read_only_memory: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "no_write_to_read_only_memory \<S>\<^sub>s\<^sub>b sb"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (Ge (CN 0 c e)) = (0 < c \<and> numbound0 e)"
subst_trm_ty_rename: fixes Y :: "tyvrs"   and t :: "trm"   and X :: "tyvrs"   and U :: "ty" assumes "Y \<sharp> t" shows "([(Y, X)] \<bullet> t)[Y \<mapsto>\<^sub>\<tau> U] = t[X \<mapsto>\<^sub>\<tau> U]"
prod_list_map_partition: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and g :: "'a \<Rightarrow> 'b" assumes "List.partition f xs = (ys, zs)" shows "prod_list (map g xs) = prod_list (map g ys) * prod_list (map g zs)"
simps_3: fixes cs :: "(Term_Class.term \<times> sterm) list" shows "frees (Sabs cs) = ffUnion (fset_of_list (map (\<lambda>(pat, rhs). frees rhs |-| frees pat) cs))"
length_zip: fixes xs :: "'a list"   and ys :: "'b list" shows "length (zip xs ys) = min (length xs) (length ys)"
LeastTr0: fixes i :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i < (LEAST l. P l)" shows "\<not> P i"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<approx> \<zero>"
iD_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> iD rt" shows "nsqn rt ip = sqn rt ip - 1"
simps_1: fixes s :: "'b stream" shows "stake 0 s = []"
sappend_assoc: fixes xs :: "[:'r:]"   and ys :: "[:'r:]"   and zs :: "[:'r:]" shows "(xs :@ ys) :@ zs = xs :@ ys :@ zs"
set_spmf_sample_set_excl: fixes Q :: "nat set"   and P :: "nat set" assumes "finite (Q - P)" shows "set_spmf (sample_set_excl Q P) = Q - P"
simps: fixes uu :: "'b"   and b :: "'a" shows "sn (Pd (uu, b)) = b"
distinct_nth_unique: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "xs ! i = xs ! j"   and "i < length xs"   and "j < length xs"   and "distinct xs" shows "i = j"
llast_LCons: fixes x :: "'b"   and xs :: "'b llist" shows "llast (LCons x xs) = (if lnull xs then x else llast xs)"