hcju/isabelleps huggingface dataset:
mapl_G_def: fixes l1 :: "'b \<Rightarrow> 'j"   and l2 :: "'d \<Rightarrow> 'k" shows "mapl_G l1 l2 = map_G l1 l2 id id id id"
id_g: shows "id_func AA \<^bsub>\<a>\<^esub> g = g"
vp_def: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" shows "CFGExit.valid_path' n as n' \<equiv> CFGExit.path n as n' \<and> CFGExit.valid_path as"
retractionI: fixes g :: "'a"   and f :: "'a" assumes "S.ide (g \<cdot>\<^sub>S f)" shows "S.retraction g"
hcomp_simps_3: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.hseq \<nu> \<mu>" shows "C.dom (\<nu> \<star>\<^sub>C \<mu>) = C.dom \<nu> \<star>\<^sub>C C.dom \<mu>"
L1_Converter: fixes z :: "nat \<Rightarrow> bit list"   and x :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Converter z x"   and "fin_make_untimed (inf_truncate z t) \<noteq> []" shows "hd (x t) = fin_make_untimed (inf_truncate z t) ! (length (fin_make_untimed (inf_truncate z t)) - 1)"
adjoint_pair_preserved_by_iso: fixes f :: "'b"   and g :: "'b"   and \<phi> :: "'b"   and f' :: "'b"   and \<psi> :: "'b"   and g' :: "'b" assumes "D.adjoint_pair f g"   and "\<guillemotleft>\<phi> : f \<Rightarrow>\<^sub>D f'\<guillemotright>"   and "D.iso \<phi>"   and "\<guillemotleft>\<psi> : g \<Rightarrow>\<^sub>D g'\<guillemotright>"   and "D.iso \<psi>" shows "D.adjoint_pair f' g'"
set_pat_vars_supp: fixes p :: "pat" shows "set (pat_vars p) = supp p"
submonoid_notin: fixes S :: "'a set" assumes "S = {x \<in> R. x \<notin> I}" shows "submonoid S R (\<cdot>) \<one>"
un: fixes x :: "real" assumes "\<exists>q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0" shows "\<exists>!q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0"
cond: fixes x :: "nat"   and n :: "nat" shows "(x < Suc n \<and> \<phi> i x \<noteq> f x) = (x < e_length (f \<triangleright> n) \<and> \<phi> i x \<down>\<noteq> e_nth (f \<triangleright> n) x)"
addfunsetD_add: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b set"   and x :: "'a"   and y :: "'a" assumes "f \<in> addfunset A M"   and "x \<in> A"   and "y \<in> A" shows "f (x + y) = f x + f y"
Nml_Inv: fixes t :: "'a local.term" assumes "Can t"   and "Nml t" shows "Nml (Inv t)"
poly_add_in_carrier: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "set (R.poly_add p1 p2) \<subseteq> carrier R"
t: shows "isTable (ExcessAt g) (vertices g) (ExcessTable g (vertices g))"
satisfies_lderiv: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "local.wf (#\<^sub>V \<AA>) \<phi>"   and "#\<^sub>V \<AA> = size x"   and "lformula \<phi>" shows "(\<AA> \<Turnstile> lderiv x \<phi>) = (CONS x \<AA> \<Turnstile> \<phi>)"
impl_inj: fixes x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and x' :: "tag"   and A' :: "agent"   and B' :: "agent"   and M' :: "msg" shows "(implem (Chan x A B M) = implem (Chan x' A' B' M')) = (x = x' \<and> A = A' \<and> B = B' \<and> M = M')"
assms: fixes S :: "'x set" assumes "open S" shows "f -` S \<inter> space M \<in> sets M"
treehead_in_multiset: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "t \<in> set bq" shows "(val t, prio t) \<in># queue_to_multiset bq"
abs_eq: fixes x :: "'a list" shows "Mapping.bulkload x = Mapping (\<lambda>k. if k < length x then Some (x ! k) else None)"
gWlsAbsDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme" shows "gWlsAbsDisj MOD \<equiv> \<forall>xs s xs' s' A. isInBar (xs, s) \<and> isInBar (xs', s') \<and> gWlsAbs MOD (xs, s) A \<and> gWlsAbs MOD (xs', s') A \<longrightarrow> xs = xs' \<and> s = s'"
remove_nth_def: fixes n :: "nat"   and xs :: "'b list" shows "remove_nth n xs \<equiv> take n xs @ drop (Suc n) xs"
le_less: fixes x :: "'b list"   and y :: "'b list" shows "subseq x y = (strict_subseq x y \<or> x = y)"
assms_1: shows "\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>y\<rparr>C"
c_nc_comp2: shows "1\<^sub>\<pi> \<sqinter> nc = (0::'a)"
add_nonpos_eq_0_iff: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subseteq># \<Empt>"   and "y \<subseteq># \<Empt>" shows "(x + y = \<Empt>) = (x = \<Empt> \<and> y = \<Empt>)"
equiv_class_eq_iff: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and x :: "'e"   and y :: "'e" assumes "equiv A r" shows "((x, y) \<in> r) = (r `` {x} = r `` {y} \<and> x \<in> A \<and> y \<in> A)"
dom_weakly_local: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = (0::'a)) = (x \<cdot> d y = (0::'a))"
topology_inverse_iff: fixes U :: "'c set \<Rightarrow> bool" shows "istopology U = (openin (topology U) = U)"
linorder: fixes ko :: "'b key_order" shows "class.linorder (ko.le ko) (ko.lt ko)"
fast_select_correct: fixes k :: "nat"   and xs :: "'a list" assumes "k < length xs" shows "fast_select k xs = select k xs"
simps_1: fixes \<alpha> :: "game"   and \<beta> :: "game" shows "Composeo (Agame \<alpha>) (Agame \<beta>) = Agame (\<alpha> ;; \<beta>)"
find_consistent_signs_R_def: fixes ps :: "real poly list" shows "find_consistent_signs_R ps = find_consistent_signs_at_roots_R (poly_f ps) ps"
simps_3: fixes uw :: "('b \<times> nat) tree"   and ux :: "'b"   and lva :: "nat"   and uy :: "('b \<times> nat) tree"   and uz :: "'b"   and lvb :: "nat"   and va :: "('b \<times> nat) tree" shows "sngl \<langle>uw, (ux, lva), \<langle>uy, (uz, lvb), va\<rangle>\<rangle> = (lvb < lva)"
A_B_step: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and a' :: "'d" assumes "Simulation_Invariant A B sim PA PB"   and "A a b"   and "PA a"   and "PB a'"   and "sim a a'" shows "\<exists>b'. B a' b' \<and> sim b b'"
carrier_vecI: fixes v :: "'a vec"   and n :: "nat" assumes "dim_vec v = n" shows "v \<in> carrier_vec n"
marked_not_white: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "white r s" shows "\<not> marked r s"
order_iff_strict: fixes b :: "'b list"   and a :: "'b list" shows "(b \<le>p a) = (b <p a \<or> a = b)"
WF_alt: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> WF(A)_v = (\<box>\<diamond>\<not> Enabled (<A>_v) \<or> \<box>\<diamond><A>_v)"
image_mset_remove1_mset_if: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and a :: "'c" shows "image_mset f (remove1_mset a M) = (if a \<in># M then remove1_mset (f a) (image_mset f M) else image_mset f M)"
domA_map_of_Some_the: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" assumes "x \<in> domA \<Gamma>" shows "map_of \<Gamma> x = Some (the (map_of \<Gamma> x))"
d_def: fixes x :: "'a" shows "d x = x \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma>"
reflp_ord_option: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp ord" shows "reflp (ord_option ord)"
fresh_atm_4: fixes a :: "coname"   and b :: "name" shows "a \<sharp> b"
Ex_list_of_length: fixes n :: "nat" shows "\<exists>xs. length xs = n"
F_def: fixes i :: "nat" shows "F i = (if i \<in> J then F' i else space M)"
aboveS_Field: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "aboveS r a \<subseteq> Field r"
subst_res_def: fixes P :: "'b assn"   and w :: "vals" shows "P\<leftarrow>w = (\<lambda>Y. P w)"
simps_4: shows "stock_measure REAL = embed_measure lborel RealVal"
WHILEI_def: shows "WHILEI \<equiv> \<lambda>I b f. REC (WHILEI_body (\<bind>) RETURN I b f)"
step_3: shows "list.set \<alpha>s \<subseteq> ON"
eq: fixes a :: "'d" assumes "a \<in> A" shows "f a = g a"
\<Delta>\<Phi>_pass2': fixes hs :: "'b tree" shows "\<Phi> (pass\<^sub>2 hs) - \<Phi> hs \<le> log 2 (real (size hs + 1))"
normF_ell2norm: fixes F :: "'a set"   and x :: "'a \<Rightarrow> complex" assumes "finite F"   and "has_ell2_norm x" shows "normF F x \<le> ell2_norm x"
span_zero: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "(0::'b) \<in> module.span scale S"
assms_4: shows "normalize n = (\<Prod>p\<in>S. p ^ f p)"
add_col_sub_row_similar: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat"   and l :: "nat"   and a :: "'b" assumes "A \<in> carrier_mat n n"   and "k < n"   and "l < n"   and "k \<noteq> l" shows "similar_mat (add_col_sub_row a k l A) A"
iffb_inv2_iff: fixes \<phi> :: "'c set \<Rightarrow> 'd set" shows "((fb\<^sub>\<F> \<circ> fb\<^sup>-\<^sub>\<F>) \<phi> = \<phi>) = Inf_pres \<phi>"
neval_S_n_1: shows "lcn \<leadsto> lcn''"
landau_symbol_axioms: shows "landau_symbol bigo bigo bigo"
map_bset_bempty: fixes f :: "'e \<Rightarrow> 'c" shows "map_bset f bempty = bempty"
fin_support_def: fixes z :: "'d"   and A :: "'c set" shows "fin_support z A = {f. finite (support z A f)}"
sum_squares_gt_zero_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (plus (times x x) (times y y)) = (x \<noteq> zero \<or> y \<noteq> zero)"
bin_rest_trunc: fixes n :: "nat"   and bin :: "int" shows "take_bit n bin div 2 = take_bit (n - 1) (bin div 2)"
le_numeral_simps_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
real_minus_mult_self_le: fixes u :: "real"   and x :: "real" shows "- (u * u) \<le> x * x"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<oplus>\<^sub>a b = b \<oplus>\<^sub>a a"
nat_less_le: fixes m :: "nat"   and n :: "nat" shows "(m < n) = (m \<le> n \<and> m \<noteq> n)"
map_cong0: fixes x :: "'c dual"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>z. z \<in> set_dual x \<Longrightarrow> f z = g z" shows "map_dual f x = map_dual g x"
\<alpha>_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "arr f"   and "arr g"   and "arr h" shows "\<a>[f, g, h] = (f \<otimes> g \<otimes> h) \<cdot> \<a>[local.dom f, local.dom g, local.dom h]"
sys_block_sizes_def: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" assumes "incidence_system point_set block_collection" shows "incidence_system.sys_block_sizes block_collection \<equiv> {int (card bl) |bl. bl \<in># block_collection}"
not_integrable_integral_eq: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<not> integrable M f" shows "integral\<^sup>L M f = (0::'b)"
packet_independent_unknown_match_tacs_2: shows "packet_independent_\<alpha> in_doubt_deny"
IH: fixes x0 :: "real \<times> real" assumes "d \<in> set (polychain_of x0 xs)"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "Counterclockwise_2D_Arbitrary.lex (snd d) x0"
step_3: obtains xs :: "(accent \<times> 'b) list" where "snd ` set xs \<subseteq> M" and "(x, v) \<in> local.lconv xs"
H_2: shows "Cauchy z \<and> Cauchy t \<and> (\<lambda>n. dist (z n) (t n)) \<longlonglongrightarrow> 0"
abc_comp_null_right: fixes A :: "abc_inst list" shows "A [+] [] = A"
IH_2: fixes w :: "nat \<Rightarrow> 'a set" assumes "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"   and "w \<Turnstile>\<^sub>n \<phi>2[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n \<phi>2"
vfind_match_rel': fixes t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and cs :: "(Term_Class.term \<times> sterm) list" assumes "t\<^sub>1 \<approx>\<^sub>e t\<^sub>2" shows "rel_option (rel_prod (fmrel erelated) (=)) (vfind_match cs t\<^sub>1) (vfind_match cs t\<^sub>2)"
not_arr_null: shows "\<not> \<F>C.arr \<F>C.null"
ws_idecl_all: fixes G :: "prog" assumes "G = tprg" shows "\<forall>(I, i)\<in>set Ifaces. ws_idecl G I (isuperIfs i)"
inf_MonoTran_def: shows "(\<sqinter>) \<equiv> map_fun Rep_MonoTran (map_fun Rep_MonoTran Abs_MonoTran) (\<sqinter>)"
final_NF: fixes a :: "'a" shows "is_final a = (a \<in> NF {(x, y). x \<rightarrow> y})"
inverse_mat_base_case_R: shows "inverts_mat (mat_of_rows_list 3 [[1 / 2, - 1 / 2, 1 / 2], [0, 1, 0], [1 / 2, - 1 / 2, - 1 / 2]]) (mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]])"
sets_measure_of_conv: fixes \<Omega> :: "'d set"   and A :: "'d set set"   and \<mu> :: "'d set \<Rightarrow> ennreal" shows "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})"
dom_char: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.dom f = (if J_C.arr f then J_C.MkIde (J_C.Dom f) else J_C.null)"
f_def: fixes M :: "complex mat"   and v :: "'a" shows "f M v = (if M \<in> carrier_mat m n then basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum v) else (0::'b))"
incseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'c" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
generate_valid_topology_some_nodes: fixes M :: "'a SecurityInvariant_configured list"   and V :: "'a set"   and E :: "('a \<times> 'a) set" shows "nodes (generate_valid_topology_some M Es \<lparr>nodes = V, edges = E\<rparr>) = V"
geodesic_segment_between_x_x_3: fixes G :: "'b set"   and x :: "'b" shows "geodesic_segment_between G x x = (G = {x})"
not_zero_bB: shows "\<not> is_zero_row_upt_k b (Suc k) B"
eq_iff_2: fixes exp :: "exp"   and var :: "Vars.var"   and expa :: "exp"   and vara :: "Vars.var" shows "(App exp var = App expa vara) = (exp = expa \<and> var = vara)"
Rep_uexpr_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> UNIV" obtains x :: "('d, 'c) uexpr" where "y = \<lbrakk>x\<rbrakk>\<^sub>e"
axioms_1: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and N :: "'c set" assumes "RModuleIso R smult M smult' T N" shows "RModuleHom R smult M smult' T"
NegNeg_D: fixes H :: "fm set"   and A :: "fm" assumes "H \<turnstile> Neg (Neg A)" shows "H \<turnstile> A"
A: shows "(\<lambda>x y. \<exists>u. u 0 = 0 \<and> (\<box> lft_rel_st rel_count) u x y) = (\<lambda>x y. y = count x)"
2: shows "z\<^sup>\<star> * (x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * z\<^sup>+ \<le> z\<^sup>\<star> * (x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * z\<^sup>\<star>"
Frct_le: fixes a :: "int"   and b :: "num"   and c :: "int"   and d :: "num" shows "(Frct (a, numeral b) \<le> Frct (c, numeral d)) = (a * numeral d \<le> c * numeral b)"
properties_for_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "mset ys = mset xs"   and "linorder.sorted less_eq ys" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = ys"
compl_sup_top: fixes x :: "'a" shows "- x + x = 1"
2: shows "limit (run\<^sub>t \<delta>' q\<^sub>0' w) = range (suffix k (run\<^sub>t \<delta>' q\<^sub>0' w))"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "inf a b \<le> a"
gbinomial_negated_upper: fixes a :: "'a"   and k :: "nat" shows "a gchoose k = (- (1::'a)) ^ k * (of_nat k - a - (1::'a) gchoose k)"
bdd_below_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq A = (\<exists>m. \<forall>x\<in>A. less_eq m x)"
case_prodI2': fixes p :: "'c \<times> 'd"   and c :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
power_le_one: fixes a :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a"   and "a \<le> (1::'a)" shows "a ^ n \<le> (1::'a)"
degree_prod_eq_sum_degree: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "\<forall>i\<in>A. f i \<noteq> 0" shows "degree (prod f A) = (\<Sum>i\<in>A. degree (f i))"
Increasing_def: fixes f :: "'d \<Rightarrow> 'e" shows "Increasing f \<equiv> \<Inter>z. Stable {s. z \<le> f s}"
check_eqv_def: fixes r :: "'b rexp"   and s :: "'b rexp" shows "check_eqv r s = (let r' = init r; s' = init s in case closure ([(r', s')], {(post r', post s')}) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False)"
convolution_finite: fixes M :: "'a measure"   and N :: "'a measure" assumes "finite_measure M"   and "finite_measure N"   and "sets N = sets borel"   and "sets M = sets borel" shows "finite_measure (M \<star> N)"
value_collect: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal" assumes "flow \<Delta>'' f" shows "value_flow \<Delta> (\<lambda>(x, y). f (Edge x y, Vertex y)) = value_flow \<Delta>'' f"
plus_def: shows "RegExp2NA.plus = (\<lambda>(q, d, f). (q, \<lambda>a s. d a s \<union> (if f s then d a q else {}), f))"
assms_2: fixes list :: "('a \<times> 'b) list"   and a :: "'a"   and b :: "'b" assumes "P list" shows "P ((a, b) # list)"
plus_mult_cpx: shows "plus_mult 1 (*) 0 (+) (a_inv cpx_rng)"
\<tau>l: shows "labels \<tau> = [hd (labels \<tau>)]"
residue_hurwitz_zeta: fixes a :: "real" assumes "0 < a" shows "residue (hurwitz_zeta a) 1 = 1"
interchange_rows_code_nth: fixes A :: "(('c, 'd) vec, 'b) vec"   and a :: "'b"   and b :: "'b" shows "($) (interchange_rows A a b) = interchange_rows_row A a b"
congruent2I': fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>y1 z1 y2 z2. \<lbrakk>(y1, z1) \<in> r1; (y2, z2) \<in> r2\<rbrakk> \<Longrightarrow> f y1 y2 = f z1 z2" shows "congruent2 r1 r2 f"
simps_4: fixes c :: "int"   and e :: "MIR.num" shows "\<beta> (NEq (CN 0 c e)) = [Neg e]"
feasible_bounded_by: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real"   and b :: "real" assumes "feasible t"   and "sound P"   and "bounded_by b P" shows "bounded_by b (t P)"
ball_subset_cball_iff: fixes a :: "'b"   and r :: "real"   and a' :: "'b"   and r' :: "real" shows "(ball a r \<subseteq> cball a' r') = (dist a a' + r \<le> r' \<or> r \<le> 0)"
list_of_dlist_cases: fixes y :: "'c list" assumes "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}" obtains x :: "'c set_dlist" where "y = list_of_dlist x"
inj_mult_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'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" assumes "class.idom times one plus zero minus uminus" shows "inj (times a) = (a \<noteq> zero)"
Y_def: shows "Y = (\<lambda>i. if i \<in> I then X i else X j)"
tm_abs_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat" assumes "x \<in>\<^sub>i range_tm e t"   and "num_params (tm_poly t) \<le> length I"   and "develops_at_within e a I" shows "\<bar>x\<bar> \<in>\<^sub>i range_tm e (tm_abs prec I a t)"
simps_5: fixes v :: "'d strand_label"   and vb :: "('e, 'f) Term.term"   and vc :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (v, delete\<langle>vb,vc\<rangle>) = (v, delete\<langle>vb,vc\<rangle>)"
Assumption_in_nodes: fixes a :: "'form" shows "(Assumption a \<in> i.R nodes) = (a \<in> set assumptions)"
invKey: fixes K :: "nat" shows "invKey (invKey K) = K"
LID\<^sub>G_def: fixes s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option" shows "LID\<^sub>G s \<equiv> \<Union> (LID\<^sub>L ` ran s)"
disj: shows "disjoint_family (\<lambda>n. {x \<in> space M. P n x} \<inter> S)"
l_new_element_get_child_nodes_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_child_nodes_axioms get_child_nodes get_child_nodes_locs \<equiv> (\<forall>ptr' new_element_ptr h h' r. ptr' \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_element_ptr h'. h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r [])"
ring_axioms: shows "ring carrier_local_ring_at add_local_ring_at mult_local_ring_at zero_local_ring_at one_local_ring_at"
rGeqT: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r2 \<le> r1" shows "[Geq \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> True"
nu_int: fixes x :: "'a" shows "\<nu> x \<le> x"
Int_in_lattice: fixes x :: "'b set"   and L :: "'b set set"   and y :: "'b set" assumes "x \<in> L"   and "y \<in> L"   and "lattice L" shows "x \<inter> y \<in> L"
defensive_strategy_Even_def: fixes p :: "'a list" shows "defensive_strategy_Even p \<equiv> SOME m. move_defensive_by_Even m p"
max_ideal_shape_aux: fixes a :: "'c \<Rightarrow> 'd"   and X :: "'c set" shows "(\<lambda>x. monomial (1::'d) (monomial 1 x) - monomial (a x) 0) ` X \<subseteq> P[X]"
cut_greaterI: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I"   and "t < x" shows "x \<in> I \<down>> t"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs H" shows "edge_rev HM a = a"
subset_dual: fixes X :: "'b set"   and Y :: "'b dual set" shows "(\<partial> ` X = Y) = (X = \<partial>\<^sup>- ` Y)"
exec_mono: fixes s :: "state"   and c :: "stmt"   and n :: "nat"   and t :: "state"   and m :: "nat" assumes "s -c-n\<rightarrow> t"   and "n \<le> m" shows "s -c-m\<rightarrow> t"
Iyx: shows "M \<le> Gromov_product_at py x x'"
unit_table_def: fixes n :: "nat" shows "unit_table n = {replicate n None}"
const_poly_dvd: fixes a :: "'b"   and b :: "'b" shows "([:a:] dvd [:b:]) = (a dvd b)"
prj_in_hom_5: shows "\<guillemotleft>\<mu>\<nu>\<pi>.Prj\<^sub>1\<^sub>0 : H\<mu>H\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<nu>.apex\<guillemotright>"
prems_7: shows "set ys \<subseteq> set (ops @ [a])"
xeq2: fixes s :: "Store"   and t :: "Store" assumes "\<forall>x. s \<equiv>[x] t"   and "\<forall>f. s@@staticLoc f = t@@staticLoc f" shows "s = t"
fail_prop: shows "target (map fst pathF || map fst (map snd pathF) || map snd (map snd pathF)) (initial M2, initial M1) = FAIL \<and> 0 < length (map fst pathF) \<and> path M2 (butlast (map fst pathF || map fst (map snd pathF))) (initial M2) \<and> path M1 (butlast (map fst pathF || map snd (map snd pathF))) (initial M1) \<and> succ M2 (last (map fst pathF)) (target (butlast (map fst pathF || map fst (map snd pathF))) (initial M2)) = {} \<and> succ M1 (last (map fst pathF)) (target (butlast (map fst pathF || map snd (map snd pathF))) (initial M1)) \<noteq> {}"