hcju/isabelleps huggingface dataset:
filterlim_at_bot_mirror: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" shows "filterlim f F at_bot = (LIM x at_top. f (- x) :> F)"
dilating_fun_injects: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and A :: "nat set" assumes "dilating_fun f r" shows "inj_on f A"
ide_prod: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "ide (hfsetcat.prod a b)"
Image_singleton_iff: fixes b :: "'d"   and r :: "('e \<times> 'd) set"   and a :: "'e" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
collapse_1: fixes llist :: "'c llist" assumes "lnull llist" shows "llist = LNil"
inject_3: fixes x41 :: "'a"   and x42 :: "('a, 'b, 'c) cmd"   and y41 :: "'a"   and y42 :: "('a, 'b, 'c) cmd" shows "(While x41 x42 = While y41 y42) = (x41 = y41 \<and> x42 = y42)"
uninst_allVars_simps_4: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and pa :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option" shows "uninst.allVars u (pa(g := p g)) g = uninst.allVars u p g"
delta_pos_neg_least_1: fixes rm :: "('d, 'a) synlr"   and rp :: "('d, 'a) synlr" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_neg \<le> dual rm"
empty_path: assumes "ns = [n]"   and "m = n" shows "thesis"
sOpPrSWls_def: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c) semDom_scheme" shows "sOpPrSWls SEM \<equiv> \<forall>delta sinp sbinp. sWlsInp SEM delta sinp \<and> sWlsBinp SEM delta sbinp \<longrightarrow> sWls SEM (stOf delta) (sOp SEM delta sinp sbinp)"
lincomb_Cons: fixes r :: "'r"   and rs :: "'r list"   and m :: "'n"   and ms :: "'n list" shows "(r # rs) \<bullet>\<star> (m # ms) = r \<star> m + rs \<bullet>\<star> ms"
inj_map_strong: fixes x :: "'b set['c]"   and xa :: "'b set['c]"   and f :: "'b \<Rightarrow> 'd"   and fa :: "'b \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> set_bset x; za \<in> set_bset xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_bset f x = map_bset fa xa" shows "x = xa"
null_sets_UN: fixes N :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>i. N i \<in> null_sets M" shows "\<Union> (range N) \<in> null_sets M"
iT_div_2: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "[n\<dots>] \<oslash> k = [n div k\<dots>]"
non_tickFree_implies_nonMt: fixes s :: "'a event list" assumes "\<not> tickFree s" shows "s \<noteq> []"
z_zm: shows "ym = zm(i \<mapsto>\<^sub>f z) \<and> fmlookup zm i = None"
cbox01_nonempty: shows "cbox (0::'c) One \<noteq> {}"
select_convs_1: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "ext \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = ext"
subset_empty: fixes A :: "'c set" shows "(A \<subseteq> {}) = (A = {})"
before_in_swap: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat"   and x :: "'a"   and y :: "'a" assumes "dist_perm xs ys"   and "Suc n < length xs" shows "x < y in swap n xs = (x < y in xs \<and> \<not> (x = xs ! n \<and> y = xs ! Suc n) \<or> x = xs ! Suc n \<and> y = xs ! n)"
nn_integral_linear: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal"   and a :: "ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. a * f x + g x \<partial>M = a * integral\<^sup>N M f + integral\<^sup>N M g"
finite_branch_nominals: fixes branch :: "(('c, 'b) fm list \<times> 'b) list" shows "finite (branch_nominals branch)"
smaps_over_un: fixes m1 :: "'a \<Rightarrow> 'b set"   and A :: "'a set"   and B :: "'b set"   and m2 :: "'a \<Rightarrow> 'b set" assumes "m1 \<in> smaps_over A B"   and "m2 \<in> smaps_over A B" shows "m1 \<union>. m2 \<in> smaps_over A B"
hyps: fixes c :: "('a, 'b) conf" assumes "se_star c ls c'" shows "states c' = {\<sigma>'. \<exists>\<sigma>\<in>states c. ce_star \<sigma> ls \<sigma>'}"
alist_of_push: fixes k :: "'c"   and p :: "'d"   and q :: "('c, 'd) pq" shows "pq.alist_of (PQ.push k p q) = (if k \<notin> set |q| then insort_key snd (k, p) (pq.alist_of q) else pq.alist_of q)"
simps_5: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (EPrim f e1 e2) \<rho> = {v. \<exists>n1 n2. VNat n1 \<in> E e1 \<rho> \<and> VNat n2 \<in> E e2 \<rho> \<and> v = VNat (f n1 n2)}"
card_remove_fset_iff: fixes y :: "'b"   and S :: "'b fset" shows "card_fset (remove_fset y S) = (if y |\<in>| S then card_fset S - 1 else card_fset S)"
8: shows "prim_E w v e \<le> - v\<^sup>T"
prems: shows "\<exists>a. i cd\<^bsup>\<pi>\<^esup>\<rightarrow> a"
trm_prm_simp_7: fixes \<pi> :: "'b prm"   and P :: "'b trm" shows "\<pi> \<cdot> Snd P = Snd (\<pi> \<cdot> P)"
emb_Liminf_llist_infinite: fixes Xs :: "'b set llist"   and Ys :: "'b set llist" assumes "emb Xs Ys"   and "\<not> lfinite Xs" shows "Liminf_llist Ys \<subseteq> Liminf_llist Xs"
H3_3: shows "AE x in M. 0 \<le> g3 x \<and> g3 x \<le> max 0 (C x - f2 x)"
hmset_of_enat_eq_\<omega>_iff: fixes n :: "enat" shows "(hmset_of_enat n = \<omega>) = (n = \<infinity>)"
halts_def: fixes p :: "(action \<times> nat) list"   and ns :: "nat list" shows "halts p ns \<equiv> {\<lambda>tp. tp = ([], <ns>)} p {\<lambda>tp. \<exists>k n l. tp = (Bk \<up> k, <n> @ Bk \<up> l)}"
splitAt_def: fixes c :: "'a"   and as :: "'a list" shows "splitAt c as \<equiv> splitAtRec c [] as"
eq2: shows "P' + Q = tensor_P (proj_psi_l 0) (proj_k 0) + matrix_sum d (\<lambda>l. tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))) R"
welldefined1: shows "list_constr (fst I) (length qs)"
tSup_empty: fixes b :: "'g" shows "tSup b {} = TNil b"
not_inftyI: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a < b"   and "b < c" shows "\<bar>b\<bar> \<noteq> \<infinity>"
has_witness_three: fixes n :: "nat"   and S :: "'a set" assumes "has n S"   and "3 \<le> n" shows "\<exists>x y z. hasw [x, y, z] S"
ln_fact_bounds_1: fixes n :: "nat" assumes "0 < n" shows "ln (2 * pi * real n) / 2 + real n * ln (real n) - real n \<le> ln (fact n)"
\<tau>red1r_Val: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and v :: "'addr val"   and xs :: "'addr val list"   and s' :: "'addr expr1 \<times> 'addr val list" shows "\<tau>red1gr uf P t h (Val v, xs) s' = (s' = (Val v, xs))"
modifies_refl: fixes vs :: "char list set"   and a :: "char list \<Rightarrow> int \<Rightarrow> int" shows "modifies vs a a"
vars1': fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space M" shows "\<forall>x\<in>free_vars e. val_type (\<sigma> x) = \<Gamma> x"
field_differentiable_at_Arcsin: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "Arcsin field_differentiable at z"
dim_def: shows "AB.dim = (LEAST n. \<exists>Aa. finite Aa \<and> card Aa = n \<and> Aa \<subseteq> mat_kernel (A * B) \<and> AB.span Aa = mat_kernel (A * B))"
delivery_has_a_cause: fixes m :: "'a \<times> 'b"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j. Broadcast m \<in> set (history j)"
add_mset_eq_add_mset: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and M' :: "'b multiset" shows "(add_mset a M = add_mset b M') = (a = b \<and> M = M' \<or> a \<noteq> b \<and> b \<in># M \<and> add_mset a (remove1_mset b M) = M')"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and 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" assumes "known_ptr = ShadowRootClass.known_ptr"   and "type_wf = ShadowRootClass.type_wf"   and "get_child_nodes = Shadow_DOM.get_child_nodes"   and "get_child_nodes_locs = Shadow_DOM.get_child_nodes_locs" shows "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf known_ptr get_child_nodes get_child_nodes_locs"
simps_2: shows "count_derangements (Suc 0) = 0"
reachable_trace: fixes q :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and p :: "nat \<times> 'c"   and r :: "(nat \<times> 'c) stream" assumes "q \<in> greachable A w V p"   and "grun A w V r q" shows "sset (gtrace r q) \<subseteq> greachable A w V p"
DataSpace_def: shows "Data.DataSpace = snd \<circ> Rep_data"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and SRel :: "('a \<times> 'a) set"   and TRel :: "('b \<times> 'b) set"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelRSTPO Enc SRel TRel"   and "(Q, R) \<in> encoding.indRelRSTPO Enc SRel TRel" shows "(P, R) \<in> encoding.indRelRSTPO Enc SRel TRel"
compatTst_def: fixes tst :: "'test" shows "compatTst tst \<equiv> \<forall>s t. s \<approx> t \<longrightarrow> tval tst s = tval tst t"
add_right_strict_mono: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V"   and \<delta> :: "V" assumes "\<alpha> \<le> \<beta>"   and "\<gamma> < \<delta>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>"   and "Ord \<delta>" shows "\<alpha> + \<gamma> < \<beta> + \<delta>"
path_component_def: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "path_component S x y \<equiv> \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
Weaken: fixes A :: "triple set"   and C' :: "triple set"   and C :: "triple set" assumes "A |\<turnstile> C'"   and "C \<subseteq> C'" shows "A |\<turnstile> C"
sort_of_atom_name: fixes a :: "name" shows "sort_of (atom a) = Sort ''name'' []"
noWhileL_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "noWhileL cl \<equiv> \<forall>c\<in>set cl. noWhile c"
nat_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "\<And>n. P n \<Longrightarrow> P (Suc n)" shows "P n"
ext_inject: fixes sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d"   and sccs' :: "'c set set"   and lowlink' :: "'c \<Rightarrow> nat option"   and tj_stack' :: "'c list"   and more' :: "'d" shows "(\<lparr>sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = \<lparr>sccs = sccs', lowlink = lowlink', tj_stack = tj_stack', \<dots> = more'\<rparr>) = (sccs = sccs' \<and> lowlink = lowlink' \<and> tj_stack = tj_stack' \<and> more = more')"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "B.par f f'"   and "B.L f = B.L f'" shows "f = f'"
sup_state_opt_trans: fixes P :: "'a prog"   and a :: "(ty list \<times> ty err list) option"   and b :: "(ty list \<times> ty err list) option"   and c :: "(ty list \<times> ty err list) option" assumes "P \<turnstile> a \<le>' b"   and "P \<turnstile> b \<le>' c" shows "P \<turnstile> a \<le>' c"
eq_dff_sym: fixes x :: "'b fract"   and y :: "'b fract" assumes "x =dff y" shows "y =dff x"
uminus_injective: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "uminus \<circ> f = uminus \<circ> g" shows "f = g"
zcount_empty: fixes a :: "'b" shows "zcount {#}\<^sub>z a = 0"
typ_of1_imp_no_loose_bvar: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts t = Some ty" shows "\<not> loose_bvar t (length Ts)"
eq_refl: fixes x :: "'a list"   and y :: "'a list" assumes "x = y" shows "subseq x y"
le_utrans_equiv_rsp_right: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and v :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_utrans t u" shows "le_utrans v t = le_utrans v u"
horizontal_composite: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<sigma> :: "'b \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'c \<Rightarrow> 'd"   and K :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation B C H K \<tau>" shows "natural_transformation A C (H \<circ> F) (K \<circ> G) (\<tau> \<circ> \<sigma>)"
subset_imp_msubset_mset_set: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B"   and "finite B" shows "mset_set A \<subseteq># mset_set B"
big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g"   and "g \<in> L F h" shows "f \<in> L F h"
comp_fun_idem: fixes x :: "'a \<times> 'b" shows "insert_mapping x \<circ> insert_mapping x = insert_mapping x"
isMinim_def: fixes A :: "'a set"   and b :: "'a" shows "r.isMinim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (b, a) \<in> r)"
act_conj_act_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<phi>) in v]"
intro: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_memberD: fixes l :: "'d strand_label"   and a :: "('e, 'f) stateful_strand_step"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" assumes "(l, a) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)" shows "\<exists>b. (l, b) \<in> set A \<and> dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, b) = (l, a)"
add_top: fixes x :: "'b" assumes "(0::'b) \<le> x" shows "x + top = top"
Conjunctive_distr_conj: fixes HC :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred"   and Q :: "'d upred" assumes "Conjunctive HC" shows "HC (P \<and> Q) = (HC P \<and> HC Q)"
sound_sum: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s + Q s)"
z_def: shows "z = (\<Sum>x | x \<in> C \<and> 0 < u x. u x *\<^sub>R x) /\<^sub>R sum u {x \<in> C. 0 < u x}"
arr_MkArr: fixes F :: "'j \<Rightarrow> 'c"   and G :: "'j \<Rightarrow> 'c"   and \<tau> :: "'j \<Rightarrow> 'c" shows "J_C.arr (J_C.MkArr F G \<tau>) = natural_transformation (\<cdot>\<^sub>J) (\<cdot>) F G \<tau>"
that: fixes E :: "'a \<Rightarrow> 'b set" assumes "A = Pi\<^sub>E I E"   and "E \<in> (\<Pi> i\<in>I. sets (M i))" shows "thesis"
add_zero_expansion_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "plus F zero_expansion = F"
chine_is_induced_map: shows "\<mu>.is_induced_map \<mu>.chine"
ex_in_conv: fixes A :: "'k set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
filterlim_at_within_If: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and G :: "'b filter"   and x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b" assumes "class.topological_space open"   and "filterlim f G (topological_space.at_within open x (A \<inter> {x. P x}))"   and "filterlim g G (topological_space.at_within open x (A \<inter> {x. \<not> P x}))" shows "LIM x topological_space.at_within open x A. if P x then f x else g x :> G"
simps: fixes x :: "nat"   and y :: "nat" shows "quo [x, y] = (let Rr = \<lambda>zs. zs ! Suc 0 * zs ! Suc (Suc 0) \<le> zs ! 0 \<and> zs ! Suc 0 \<noteq> 0 in Maxr Rr [x, y] x)"
max_pos_lit_dominates_cl: fixes L :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list"   and C :: "'a literal set"   and t :: "'a trm"   and s :: "'a trm"   and L' :: "'a literal"   and I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool" assumes "maximal_literal (subst_lit L \<sigma>) (subst_cl C \<sigma>)"   and "orient_lit_inst L t s pos \<sigma>"   and "L' \<in> C - {L}"   and "\<not> equivalent_eq_exists t s C I \<sigma> L"   and "vars_of_lit (subst_lit L \<sigma>) = {}"   and "vars_of_lit (subst_lit L' \<sigma>) = {}"   and "fo_interpretation I" shows "(subst_lit L' \<sigma>, subst_lit L \<sigma>) \<in> lit_ord"
UNIV_eq_I: fixes A :: "'g set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
not_in_lift: fixes var :: "nat"   and p :: "real mpoly"   and z :: "nat" assumes "var \<notin> vars p" shows "var + z \<notin> vars (liftPoly 0 z p)"
rQuots_append: fixes v :: "'b list"   and w :: "'b list"   and A :: "'b list set" shows "(v \<in> rQuots w A) = (v @ rev w \<in> A)"
hd_append: fixes xs :: "'b list"   and ys :: "'b list" shows "hd (xs \<cdot> ys) = (if xs = \<epsilon> then hd ys else hd xs)"
compress_normalize_primitive_hasdisc: fixes disc2 :: "'c \<Rightarrow> bool"   and m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr" assumes "\<not> has_disc disc2 m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc2 (C a)"   and "normalized_nnf_match m"   and "compress_normalize_primitive (disc, sel) C f m = Some m'" shows "normalized_nnf_match m' \<and> \<not> has_disc disc2 m'"
indep: shows "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
it_ext: fixes x :: "'a" shows "x \<le> x\<^sup>\<infinity>"
Rep_uint8_inject: fixes x :: "uint8"   and y :: "uint8" shows "(Rep_uint8 x = Rep_uint8 y) = (x = y)"
diverge_def: fixes e :: "exp" shows "diverge e \<equiv> \<forall>e'. e \<longrightarrow>* e' \<longrightarrow> (\<exists>e''. e' \<longrightarrow> e'')"
agree_comm: fixes A :: "(real, 'a) vec \<times> (real, 'a) vec"   and B :: "(real, 'a) vec \<times> (real, 'a) vec"   and V :: "('a + 'a) set" assumes "Vagree A B V" shows "Vagree B A V"
hyps_1: shows "((e\<^sub>0, s\<^sub>0), e\<^sub>1, s\<^sub>1) \<in> Red P (E(V \<mapsto> T))"
Inf_union_distrib: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (A \<union> B) = inf (Inf A) (Inf B)"
second_def: fixes s :: "nat \<Rightarrow> 'b" shows "second s \<equiv> s 1"
finsum_empty: fixes f :: "'c \<Rightarrow> 'a" shows "finsum {} f = \<zero>"
assms_5: shows "wf\<^sub>e\<^sub>s\<^sub>t {} \<A>\<^sub>d"
hyps_3: assumes "jm \<in> DeclConcepts.imethds G J sig" shows "\<exists>im\<in>DeclConcepts.imethds G SI sig. is_static im = is_static jm \<and> accmodi im = accmodi jm \<and> G\<turnstile>resTy im\<preceq>resTy jm"
finite_subs: fixes y :: "(nat \<times> form) list" shows "finite {w. \<not> is_axiom (s_of_ns y) \<and> w \<in> set (subs y)}"
treewidth_cards_treewidth: shows "\<exists>T bag. TreeDecomposition G T bag \<and> treewidth = TreeDecomposition.width T bag"
ENR_imp_ANR: fixes S :: "'a set" assumes "ENR S" shows "ANR S"
sin_int_times_real: fixes m :: "int"   and x :: "real" shows "sin (of_int m * of_real x) = of_real (sin (real_of_int m * x))"
strand_sem_split_2: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>c \<theta>"
subst_poly_def: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and p :: "linear_poly" shows "subst_poly \<sigma> p = (\<Sum>x\<in>vars p. coeff p x *R \<sigma> x)"
analyzed_closed_mod_timpls_is_analyzed_closed_timpl_closure_set_aux1: fixes f :: "'fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term list"   and K :: "(('fun, 'atom, 'sets) prot_fun, nat) Term.term list"   and R :: "nat list"   and i :: "nat"   and M :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term set"   and TI :: "('sets set \<times> 'sets set) set"   and S :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term list" assumes "arity\<^sub>f f = length T"   and "0 < arity\<^sub>f f"   and "Ana\<^sub>f f = (K, R)"   and "i < length R"   and "timpl_closure_set M TI \<turnstile>\<^sub>c T ! (R ! i)"   and "\<langle>f T\<rangle>\<^sub>t \<in> M"   and "\<langle>f S\<rangle>\<^sub>t \<in> timpl_closure \<langle>f T\<rangle>\<^sub>t TI" shows "timpl_closure_set M TI \<turnstile>\<^sub>c S ! (R ! i)"
ind_in_set_bij: shows "bij_betw (ind_in_set vars0) ({0..<length dims} \<inter> vars0) {0..<card ({0..<length dims} \<inter> vars0)}"
rev_iuntilI: fixes t :: "nat"   and I :: "nat set"   and Q :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "t \<in> I"   and "Q t"   and "\<And>t'. t' \<in> I \<down>< t \<Longrightarrow> P t'" shows "P t'. t' \<U> t I. Q t"
ivl_disj_un_two_3: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..<u} = {l..<u}"
\<psi>_char_3: shows "\<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (\<psi> \<otimes>\<^sub>D \<psi>)"
ivp_2: fixes s :: "'a" assumes "s \<in> S" shows "\<phi> 0 s = s"
coplanar_def: fixes S :: "'b set" shows "coplanar S \<equiv> \<exists>u v w. S \<subseteq> affine hull {u, v, w}"
eq: fixes x :: "'a" assumes "x \<in> G" shows "A x = B x"
uDQ: shows "unitary (wp D (Q \<circ> \<phi>))"
h13: shows "st_in t = voice_com \<and> stop t = [stop_vc] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state"
prob_grw_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real" shows "prob_grw p M geom_proc u d init \<equiv> infinite_coin_toss_space p M \<and> prob_grw_axioms geom_proc u d init"
bind_Ok: fixes x :: "('k, 'l) result" shows "x \<bind> Ok = x"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "i < nr"   and "j < nc" shows "Matrix.mat nr nc f $$ (i, j) = f (i, j)"
eval_single: fixes x :: "'a" shows "eval (single x) = (=) x"
nxdn: shows "isnpolyh ((shift1 ^^ (Reflected_Multivariate_Polynomial.degree s - n)) (1)\<^sub>p) 0"
Q1'_def: shows "Q1' = (\<lambda>t t' X. \<forall>x\<in>fv t \<union> fv t' - set X. \<exists>a. \<Gamma> (Var x) = Var a)"
Nil_sp: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel subs" shows "SubRel.subpath g v1 [] v2 subs = (sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> (v1 = v2 \<or> (v1, v2) \<in> subs))"