hcju/isabelleps huggingface dataset:
1: shows "F \<circ> \<chi>.A.map = Fa.map"
runit'_simps_1: fixes f :: "'b" assumes "D.ide f" shows "D.arr \<r>\<^sub>D\<^sup>-\<^sup>1[f]"
charts_submanifold_def: fixes S :: "'a set" shows "src.charts_submanifold S = restrict_chart S ` charts1"
in_obs_valid_1: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs n S" shows "valid_node n"
det_transpose: fixes A :: "(('a, 'b) vec, 'b) vec" shows "det (Finite_Cartesian_Product.transpose A) = det A"
strict_uv_refine_lemma_v: fixes X :: "'b program set"   and F :: "'b program"   and G :: "'b program" assumes "strict_uv_prop X" shows "(\<forall>H. F ok H \<and> G ok H \<and> F \<squnion> H \<in> welldef \<and> F \<squnion> H \<in> X \<longrightarrow> G \<squnion> H \<in> X) = (F \<in> welldef \<inter> X \<longrightarrow> G \<in> X)"
src_closed: fixes f :: "'a" assumes "B.arr f \<and> src\<^sub>B f = a \<and> trg\<^sub>B f = a" shows "B.arr (src\<^sub>B f) \<and> src\<^sub>B (src\<^sub>B f) = a \<and> trg\<^sub>B (src\<^sub>B f) = a"
RELATESI: fixes R :: "('a \<times> 'b) set" shows "RELATES R"
pcr_cr_eq: shows "transaction_merkle.pcr_transaction\<^sub>m = transaction_merkle.cr_transaction\<^sub>m"
dvd_div_eq_cancel: 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 a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "divide a c = divide b c"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "a = b"
polytope_def: fixes S :: "'c set" shows "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
map_scale_eq_mult_scalar: fixes c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> p = MPoly_Type_Class.punit.mult_scalar (monomial c (0::'b)) p"
list_ex_rec_def: shows "list_ex_rec \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. False) (\<lambda>a l la P. P a \<or> la P) uua uu"
count_remdups_mset_If: fixes A :: "'b multiset"   and a :: "'b" shows "count (remdups_mset A) a = (if a \<in># A then 1 else 0)"
wf_trm_subst_rangeD: fixes \<theta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and x :: "'b" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"
vimage2p_comp: fixes f1 :: "'j \<Rightarrow> 'e"   and f2 :: "'h \<Rightarrow> 'j"   and g1 :: "'k \<Rightarrow> 'f"   and g2 :: "'i \<Rightarrow> 'k" shows "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
primeideal_vimage: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and P :: "'e set" assumes "ring_hom_ring R S h"   and "cring R"   and "primeideal P S" shows "primeideal {r \<in> carrier R. h r \<in> P} R"
induction_axioms_into_Induction_ax: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> induction_axioms" shows "\<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<in> Induction_ax"
poly_to_row_to_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and ts :: "'t list" assumes "keys p \<subseteq> set ts" shows "row_to_poly ts (poly_to_row ts p) = p"
finite_fun_UNIVD1: assumes "finite UNIV"   and "CARD('b) \<noteq> Suc 0" shows "finite UNIV"
Tn_meas: fixes n :: "nat" shows "I.random_variable natural_extension_measure (natural_extension_map ^^ n)"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'c, 'opSym, 'var, 'a, 'b, 'd) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
that: fixes B :: "'a multiset" assumes "A = add_mset a B" shows "thesis"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<star> f = n \<and> f \<star> n = n"
add_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x + z \<le> y + z"
ket_pre_dim: shows "ket_pre \<in> carrier_vec N"
CAND_OrR1_elim: fixes a :: "coname"   and b :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:OrR1 <b>.M a \<in> \<parallel><B>\<parallel>"   and "<a>:OrR1 <b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<exists>B1 B2. B = B1 OR B2 \<and> <b>:M \<in> \<parallel><B1>\<parallel>"
rtrancl_image_advance: fixes q :: "'h"   and R :: "('h \<times> 'h) set"   and Q0 :: "'h set"   and x :: "'h" assumes "q \<in> R\<^sup>* `` Q0"   and "(q, x) \<in> R" shows "x \<in> R\<^sup>* `` Q0"
less_imp_not_eq2: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<prec>\<^sub>p y" shows "(y = x) = False"
ex_Inf: fixes A :: "'b set" assumes "CompleteLattice.class.complete_lattice TYPE('b)" shows "\<exists>inf. is_Inf A inf"
Keys_insert: fixes a :: "'a \<Rightarrow>\<^sub>0 'b"   and A :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "Keys (insert a A) = keys a \<union> Keys A"
last_step_of_path: fixes q :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and q' :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "q \<in> \<P> k u"   and "q = q' @ [t]" shows "\<exists>k' u'. indexlt k' u' k u \<and> q \<in> \<P> k' (Suc u') \<and> charslength q' = k' \<and> t \<in> \<Z> k' (Suc u')"
distinct_rev: fixes xs :: "'c list" shows "distinct (rev xs) = distinct xs"
closure_dyadic_rationals_in_convex_set: fixes S :: "'b set" assumes "convex S"   and "interior S \<noteq> {}" shows "closure (S \<inter> (\<Union>k. \<Union>f\<in>Basis \<rightarrow> \<int>. {\<Sum>i\<in>Basis. (f i / 2 ^ k) *\<^sub>R i})) = closure S"
word_sp_01: shows "word_succ (- 1) = 0 \<and> word_succ 0 = 1 \<and> word_pred 0 = - 1 \<and> word_pred 1 = 0"
card_set: fixes xs :: "'b list" shows "card (set xs) = length (remdups xs)"
dWWQk: shows "WS (W Q k) \<in> carrier_mat d d"
ord_option_simps_3: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" shows "ord_option ord (Some x) (Some y) = ord x y"
map_simp: fixes f :: "'a" assumes "arr f" shows "I.map f = \<I>"
abs_cases: fixes a :: "'a" obtains "a = (0::'a)"   | "(0::'a) < \<bar>a\<bar>"
eigenvector_hom_rev: fixes A :: "real mat"   and n :: "nat"   and v :: "real Matrix.vec"   and ev :: "real" assumes "A \<in> carrier_mat n n"   and "eigenvector (of_real_hom.mat_hom A) (of_real_hom.vec_hom v) (of_real ev)" shows "eigenvector A v ev"
left_null_space_orthogonal_complement_col_space: fixes A :: "((complex, 'b) vec, 'a) vec" shows "left_null_space A = complex_matrix.orthogonal_complement (col_space (\<chi>i j. cnj_class.cnj (A $ i $ j)))"
left_adjoint_extends_to_adjoint_pair: fixes f :: "'a arrow_of_spans_data" assumes "is_left_adjoint f" shows "adjoint_pair f f\<^sup>*"
eval_monom1: fixes s :: "'c" assumes "s \<in> carrier S" shows "eval R S h s (monom P \<one> 1) = s"
msteps_msteps_stateless: fixes w :: "(char list \<times> 'a list) prefix"   and st :: "'a mstate" shows "fst (msteps w st) = msteps_stateless w st"
ProcRec1: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and P :: "'d \<Rightarrow> 'f set"   and p :: "'e"   and Q :: "'d \<Rightarrow> 'f set"   and A :: "'d \<Rightarrow> 'f set"   and F :: "'g set" assumes "\<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>Z. {(P Z, p, Q Z, A Z)})\<turnstile>\<^bsub>/F \<^esub>(P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) Call p (Q Z),(A Z)"
ThinChamberComplexManyFoldings_axioms: shows "ThinChamberComplexManyFoldings X C0"
bind_assoc': fixes m :: "'d \<Rightarrow> ('g \<times> 'd) option"   and k :: "'d \<Rightarrow> ('f \<times> 'd) option"   and h :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "(m ;- k) ;- h = m ;- k ;- h"
lottery_prob_alts: fixes p :: "'alt pmf" assumes "p \<in> lotteries" shows "lottery_prob p alts = 1"
asBinp_pickBinp: fixes binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" assumes "goodBinp binp" shows "asBinp (pickBinp binp) = binp"
has_integral_component_lbound: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and B :: "real"   and k :: "'b" assumes "(f has_integral i) (cbox a b)"   and "\<forall>x\<in>cbox a b. B \<le> f x \<bullet> k"   and "k \<in> Basis" shows "B * content (cbox a b) \<le> i \<bullet> k"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<nu> \<pi>"
mvt: shows "f (a + u) - f a = (\<Sum>i\<in>Basis. (J (a + t i *\<^sub>R u) u \<bullet> i) *\<^sub>R i)"
inDed_mono': fixes x :: "formula list"   and y :: "formula list" assumes "set x \<subseteq> set y"   and "x \<in> deductions CutFreePC" shows "y \<in> deductions CutFreePC"
divide_nonneg_nonpos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq y zero" shows "less_eq (divide x y) zero"
inPorts_fset_of: fixes n :: "('form, 'rule) graph_node" shows "inPorts n = fset_from_list (inPorts' n)"
dep_imp_supset_circuit: fixes X :: "'a set" assumes "X \<subseteq> \<E>"   and "\<not> indep_in \<E> X" shows "\<exists>C. \<E>.circuit C \<and> C \<subseteq> X"
rel_distinct_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y2 :: "'c" shows "\<not> rel_option R (Some y2) None"
exe_consistent_length_tcsigs_imp_consistent_length_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_ars_conds arss"   and "exe_consistent_length_tcsigs arss" shows "consistent_length_tcsigs (translate_ars arss)"
submonoid_def: fixes N :: "'b set"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b" shows "submonoid N M composition unit \<equiv> Group_Theory.monoid M composition unit \<and> submonoid_axioms N M composition unit"
natural_transformation_axioms: shows "natural_transformation C C (Adj.FG.map \<circ> Adj.FG.map) Adj.FG.map (Adj.FG.map \<circ> \<epsilon>)"
additive_subgroupI: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "subgroup H (add_monoid G)" shows "additive_subgroup H G"
SFAccInitReds: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and v' :: "val"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and sfs :: "sfields"   and i :: "init_state"   and v :: "val" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. shp s D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,s,False\<rangle> \<rightarrow>* \<langle>Val v',s',b'\<rangle>"   and "shp s' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s,False\<rangle> \<rightarrow>* \<langle>Val v,s',False\<rangle>"
Says_Tgs_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and X :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X\<rbrace>) \<in> set evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK authK' servK evs"
reduced_word_for_genby_sym_arg_min: fixes A :: "'b set"   and a :: "'b" assumes "order.greater_eq A (uminus ` A)"   and "a \<in> \<langle>A\<rangle>" shows "reduced_word_for A a (arg_min length (word_for A a))"
IH: shows "vars_partitioned (subst_cls_lists Cs (renamings_apart Cs))"
list_ex_simps_2: fixes P :: "'e \<Rightarrow> bool" shows "list_ex P [] = False"
wf_irrefl: fixes r :: "('e \<times> 'e) set"   and a :: "'e" assumes "wf r" obtains "(a, a) \<notin> r"
alw_InvIncomingInfoUpright: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvIncomingInfoUpright) s"
insertE: fixes a :: "'d"   and b :: "'d"   and A :: "'d set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
C_2: shows "rep_Bonk_Schramm_extension y \<in> extended_distance_set"
S7: shows "c_cons x u = list_to_nat (x # ls)"
fdemodalisation22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(antirange_semiring.bdia (\<cdot>) ar x y \<le> r z) = (r y \<cdot> (x \<cdot> ar z) = \<bottom>)"
open_ereal_greaterThan: fixes a :: "ereal" shows "open {a<..}"
elapsed_eq: fixes u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a"   and c :: "'c" assumes "eq_elapsed u u'"   and "c \<in> \<X>"   and "(0::'a) \<le> u' c - u c" shows "elapsed u u' = u' c - u c"
sl_impl: shows "correct_subseqs_foldr_impl (\<lambda>v. map_prod (mul_const v) ((#) v)) sl_impl sli"
basis_orthogonal2: fixes xs :: "('a, 'b) vec list" shows "length (Gram_Schmidt2 xs) = length xs \<and> span (set (Gram_Schmidt2 xs)) = span (set xs) \<and> pairwise orthogonal (set (Gram_Schmidt2 xs))"
integ_c: fixes k :: "nat" assumes "k \<le> n" shows "complex_integrable M (c k)"
span_neg: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> ideal S" shows "- x \<in> ideal S"
more_than_one_mset_mset_diff: fixes a :: "'c"   and M :: "'c multiset" assumes "a \<in># M - {#a#}" shows "set_mset (M - {#a#}) = set_mset M"
ln_mult: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "ln (x * y) = ln x + ln y"
elims: fixes x :: "('c \<times> 'd) set"   and y :: "('c \<times> 'd) tree" assumes "treap_of x = y" obtains KP :: "('c \<times> 'd) set" where "x = KP" and "y = (if infinite KP \<or> KP = {} then \<langle>\<rangle> else let m = arg_min_on snd KP; L = {p \<in> KP. fst p < fst m}; R = {p \<in> KP. fst m < fst p} in \<langle>treap_of L, m, treap_of R\<rangle>)"
simps_3: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and kx :: "'a"   and vx :: "'d"   and xs :: "('a \<times> 'd) list" shows "lex_ord_pair f ((kx, vx) # xs) [] = (let aux = f kx vx (0::'e) in if aux = Some Eq then lex_ord_pair f xs [] else aux)"
evaln_elim_cases_25: fixes G :: "prog"   and s :: "st"   and T :: "ty"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (New T[e])\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains s1 :: "state" and i' :: "val" and s2 :: "state" and a :: "loc" where "v = In1 (Addr a)" and "G\<turnstile>Norm s \<midarrow>init_comp_ty T\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e-\<succ>i'\<midarrow>n\<rightarrow> s2" and "G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc Arr T (the_Intg i')\<succ>a\<rightarrow> s'"
lt_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" shows "lt p = (if p = 0 then min_term else ord_term_lin.Max (keys p))"
even_square_cong_4_int: fixes x :: "int" assumes "even x" shows "[x\<^sup>2 = 0] (mod 4)"
inf_le1: fixes x :: "'b upred"   and y :: "'b upred" shows "x \<sqsubseteq> (x \<and> y)"
Carmichael_number_iff': fixes n :: "nat" shows "Carmichael_number n = (1 < n \<and> size (prime_factorization n) \<noteq> 1 \<and> (\<forall>p\<in>#prime_factorization n. count (prime_factorization n) p = 1 \<and> p - 1 dvd n - 1))"
piecewise_C1_differentiable_compose: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on f ` S"   and "\<And>x. finite (S \<inter> f -` {x})" shows "g \<circ> f piecewise_C1_differentiable_on S"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<squnion>1 (a \<squnion>1 c) = a \<squnion>1 (b \<squnion>1 c)"
P''RelQ'': shows "(\<Psi>, P'', Q'') \<in> Rel"
stable_def: fixes p :: "('a, 'b, 'c) configuration \<Rightarrow> bool" shows "stable p \<equiv> \<forall>c. p c \<longrightarrow> (\<forall>t c'. trace c t c' \<longrightarrow> p c')"
monotone_lset: shows "monotone (\<sqsubseteq>) (\<subseteq>) lset"
mset_factorsD_1: fixes F :: "'a multiset"   and p :: "'a"   and f :: "'a" assumes "mset_factors F p"   and "f \<in># F" shows "irreducible f"
whisker_left: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.seq \<nu> \<mu>" shows "f \<circ> \<nu> \<bullet> \<mu> = (f \<circ> \<nu>) \<bullet> (f \<circ> \<mu>)"
v1_notin_S: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" assumes "Separation G v0 v1 S" shows "v1 \<notin> S"
paths_are_winning_for_one_player: fixes P :: "'a llist"   and p :: "Player" assumes "valid_path P" shows "winning_path p P = (\<not> winning_path p** P)"
prems_1: shows "depth P + depth Q \<le> Suc n"
Diff_disjoint: fixes A :: "'h set"   and B :: "'h set" shows "A \<inter> (B - A) = {}"
CollectI: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a" shows "a \<in> {x. P x}"
1: fixes a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "a \<in> set bs"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" shows "rword (spp_of a) (spp_of b)"
normal_iff_subgroup: fixes N :: "('c \<Rightarrow> 'a) set" shows "N \<lhd> DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I = subgroup N (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"
not_bot: fixes A :: "'a Predicate.pred" assumes "A \<noteq> bot" obtains x :: "'a" where "pred.eval A x"
function_to_finmap_axioms: fixes n :: "nat" shows "function_to_finmap (J n) Utn (from_nat_into (\<Union> (range J)))"
bnd: shows "AE x in M. \<forall>i. u i x \<le> w x"
mat_of_row_carrier_2: fixes y :: "'a Matrix.vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat (Suc 0) n"
D'_def: shows "D' = (\<lambda>x. - 1 / (2 * x\<^sup>2 * (x + 1)\<^sup>2))"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<mu> \<nu>"
finite_subset_induct': fixes F :: "'d set"   and A :: "'d set"   and P :: "'d set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
UNIV_unitary: fixes a :: "'a" shows "UNIV = {a}"
in_sccsE: fixes c :: "('a, 'b) pre_digraph" assumes "c \<in> sccs" obtains "induced_subgraph c G" and "strongly_connected c" and "\<nexists>d. induced_subgraph d G \<and> strongly_connected d \<and> verts c \<subset> verts d"
uniformly_convergent_on: fixes A :: "'c set"   and g :: "nat \<Rightarrow> 'c \<Rightarrow> 'b" assumes "uniformly_convergent_on A g" shows "uniformly_convergent_on A (\<lambda>x y. ((- 1) ^ Suc n / fact n) *\<^sub>R g x y)"
fps_inv: fixes a :: "'a fps" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 \<noteq> (0::'a)" shows "fps_inv a oo a = fps_X"
ex_no_call: fixes S :: "'c rule list set" assumes "finite S" shows "\<exists>c. \<forall>rs\<in>S. no_call_to c rs"
sub: shows "set (ap gs bs (ps -- sps) hs data') \<subseteq> set (ab gs bs hs data') \<times> (set gs \<union> set (ab gs bs hs data'))"
lleq_dichotomy: fixes x :: "char list"   and y :: "char list" shows "lleq_charlist x y \<or> lleq_charlist y x"
isGlbD1: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isGlb R S x" shows "x <=* S"
nonneg_mat_mult: fixes A :: "'a mat"   and B :: "'a mat"   and nr :: "nat"   and n :: "nat"   and nc :: "nat" assumes "nonneg_mat A"   and "nonneg_mat B"   and "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "nonneg_mat (A * B)"
UN_empty: fixes B :: "'b \<Rightarrow> 'a set" shows "\<Union> (B ` {}) = {}"
subsets_1: shows "{m. coeff p m \<noteq> (0::'a)} \<subseteq> S"
fmfilter_empty: fixes P :: "'c \<Rightarrow> bool" shows "fmfilter P fmempty = fmempty"
path_\<mu>_def: shows "path_\<mu> = (INF s\<in>{p. apath u p v}. ereal (awalk_cost f s))"
scale_shift_ms_aux_conv_mslmap: fixes x :: "'a \<times> real" shows "scale_shift_ms_aux x = mslmap (map_prod ((*) (fst x)) ((+) (snd x)))"
DERIV_neg_imp_decreasing_open: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> y < 0"   and "continuous_on {a..b} f" shows "f b < f a"
rtrancl_consistent_rtrancl_on: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and F :: "'c set" assumes "(a, b) \<in> r\<^sup>*"   and "a \<in> F"   and "b \<in> F"   and "\<And>a b. \<lbrakk>a \<in> F; (a, b) \<in> r\<rbrakk> \<Longrightarrow> b \<in> F" shows "(a, b) \<in> rtrancl_on F r"
is_Arg_Arg: fixes z :: "complex" assumes "z \<noteq> 0" shows "is_Arg z (Arg z)"
splits_wrt_def: fixes pqs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "splits_wrt pqs T F = (cone_decomp T (fst pqs @ snd pqs) \<and> (\<forall>hU\<in>set (fst pqs). cone hU \<subseteq> ideal F \<inter> P[X]) \<and> (\<forall>(h, U)\<in>set (snd pqs). cone (h, U) \<subseteq> P[X] \<and> cone (h, U) \<inter> ideal F = {0}))"
diff_add: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># b" shows "b - a + a = b"
liveVal'I: fixes g :: "'g"   and v :: "'val" assumes "step.liveVal g v" obtains vs :: "'val list" where "step.liveVal' g (v # vs)"
r_alt_def'': fixes \<chi> :: "'a ltl"   and i :: "nat"   and q :: "'a ltl_prop_equiv_quotient" assumes "\<chi> \<in> \<^bold>G \<phi>" shows "the (snd (fst (r i)) \<chi>) q = semi_mojmir_def.state_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w q i"
y2: shows "(\<Union>x\<in>R `` {y}. {R `` {x}}) = {R `` {y}}"
sylvester_mat_sub_def: fixes m :: "nat"   and n :: "nat"   and p :: "'c poly"   and q :: "'c poly" shows "sylvester_mat_sub m n p q \<equiv> mat (m + n) (m + n) (\<lambda>(i, j). if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'c) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'c))"
simps_2: fixes ao :: "'a option"   and uv :: "'a list" shows "proper_interval_Compl_set_aux ao [] uv = False"
open_cover_of_subset_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" shows "open_cover_of_subset X is_open U I C \<equiv> topological_space X is_open \<and> cover_of_subset X U I C \<and> open_cover_of_subset_axioms is_open I C"
arr_char: fixes f :: "'a" shows "Fop.Bop.arr f = Cop.arr f"
gcd_unique: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(dvd.dvd times d a \<and> dvd.dvd times d b \<and> normalize d = d \<and> (\<forall>e. dvd.dvd times e a \<and> dvd.dvd times e b \<longrightarrow> dvd.dvd times e d)) = (d = gcd a b)"
neg_le_minus_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "c < 0" shows "(a \<le> - (b /\<^sub>R c)) = (- b \<le> c *\<^sub>R a)"