hcju/isabelleps huggingface dataset:
implmt_ObjectE: fixes G :: "prog"   and I :: "qtname"   and R :: "bool" assumes "G\<turnstile>Object\<leadsto>I" shows "R"
iT_add_5: fixes i :: "nat"   and k :: "nat" shows "[\<dots>i] \<oplus> k = [k\<dots>,i]"
alphaAbs_imp_asAbs_equal: fixes qA :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and qB :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs qA"   and "qA $= qB" shows "asAbs qA = asAbs qB"
two_power_fact_le_fact: fixes n :: "nat"   and k :: "nat" assumes "1 \<le> n" shows "(2::'a) ^ k * fact n \<le> fact (n + k)"
in_fSpan_obtain_same_length_coeffs: fixes n :: "'v"   and ms :: "'v list" assumes "n \<in> R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) ms" shows "\<exists>rs. set rs \<subseteq> UNIV \<and> length rs = length ms \<and> n = rs \<bullet>\<sharp>\<cdot> ms"
consistent_context_def: fixes \<Gamma> :: "'a constr list" shows "consistent_context \<Gamma> \<equiv> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<noteq> {}"
axioms_2: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext_singleton ext" shows "ext_singleton_axioms ext"
Invariants_subset_tail_algebra: shows "sets Invariants \<subseteq> local.P.tail_events (\<lambda>i. sets.pullback_algebra M (\<lambda>f. f i) (space P))"
hyps_C_sv: fixes b :: "contour"   and \<beta>' :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and c :: "call" assumes "b \<in> ran \<beta>'"   and "\<forall>b'\<in>ran \<beta>'. b' \<le> b"   and "\<forall>b'\<in>contours_in_ve ve. b' \<le> b" shows "single_valued (evalC\<cdot>(Discr (c, \<beta>', ve, b)))"
fixp_lfp_parametric_eq: fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and G :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)"   and "\<And>x. lfp.mono_body (\<lambda>f. G f x)"   and "rel_fun (rel_fun A (=)) (rel_fun A (=)) F G" shows "rel_fun A (=) (lfp.fixp_fun F) (lfp.fixp_fun G)"
reachable_subgraph: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and E' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and a :: "'c"   and b :: "'c" assumes "Subgraph_Start E E'"   and "Graph_Start_Defs.reachable E s\<^sub>0 a"   and "E'\<^sup>*\<^sup>* a b" shows "Graph_Start_Defs.reachable E s\<^sub>0 b"
ie4: shows "te q (i + 1) \<le> te p (i + 1)"
reachable_refl: fixes v :: "'a" assumes "v \<in> verts (G \<restriction> {v. u \<rightarrow>\<^sup>* v})" shows "v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v"
ord_pp_strict_def: fixes cmp_term :: "'a nat_term_order"   and s :: "'a"   and t :: "'a" shows "ord_pp_strict_punit cmp_term s t = lt_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))"
bounded_linear_mult_left: fixes y :: "'b" shows "bounded_linear (\<lambda>x. x * y)"
sees_method_compPD: fixes f :: "staticb \<Rightarrow> 'b \<Rightarrow> 'a"   and P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and fm :: "'a"   and D :: "char list" assumes "compP f P \<turnstile> C sees M, b :  Ts\<rightarrow>T = fm in D" shows "\<exists>m. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D \<and> f b m = fm"
select_convs_11: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getC \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getC"
prems_4: shows "x \<in> standard_simplex (Suc p)"
vec_def: fixes n :: "nat"   and x :: "'a list" shows "vec n x = (length x = n)"
baIA: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and \<phi> :: "'b \<Rightarrow> bool"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b" assumes "\<exists>i\<ge>n. active c (t i)"   and "\<phi> (tCMP c (t (dynamic_component.nxtAct active c t n)))" shows "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)"
run_get_state: fixes f :: "'s \<Rightarrow> ('s, 'm) stateT"   and s :: "'s" shows "run_state (local.get_state f) s = run_state (f s) s"
map_component_inj: fixes f :: "nat \<Rightarrow> nat"   and u :: "'t"   and v :: "'t" assumes "inj f"   and "map_component f u = map_component f v" shows "u = v"
mult_le_cancel_right: fixes 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 abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a c) (times b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
vsv_Limit_vrange_in_VsetI: fixes r :: "V"   and \<alpha> :: "V" assumes "vsv r"   and "Limit \<alpha>"   and "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vfinite (\<D>\<^sub>\<circ> r)" shows "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
dvd_div_div_eq_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "c \<noteq> zero"   and "dvd.dvd times a b"   and "dvd.dvd times c d" shows "(divide b a = divide d c) = (times b c = times a d)"
finite_set_decode: fixes n :: "nat" shows "finite (set_decode n)"
neg_antitone: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y"   and "x \<le> y" shows "! y \<le> ! x"
convergent_prod_imp_LIMSEQ: fixes f :: "nat \<Rightarrow> 'a" assumes "convergent_prod f" shows "f \<longlonglongrightarrow> (1::'a)"
mult2_alt_add: fixes X1 :: "'b multiset"   and Y1 :: "'b multiset"   and b1 :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and X2 :: "'b multiset"   and Y2 :: "'b multiset"   and b2 :: "bool" assumes "(X1, Y1) \<in> mult2_alt b1 ns s"   and "(X2, Y2) \<in> mult2_alt b2 ns s" shows "(X1 + X2, Y1 + Y2) \<in> mult2_alt (b1 \<and> b2) ns s"
fun_star_unfoldl_eq: fixes f :: "'b nd_fun" shows "1 \<squnion> f \<cdot> qstar f = qstar f"
chain_union_closed: fixes C :: "'b set set"   and f :: "nat \<Rightarrow> 'b set" assumes "finite_char C"   and "is_chain f"   and "\<forall>n. f n \<in> C" shows "\<Union> (range f) \<in> C"
rel_option_map1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and x :: "'e option"   and y :: "'d option" shows "rel_option R (map_option f x) y = rel_option (\<lambda>x. R (f x)) x y"
subint_widen_imethds: fixes I :: "qtname"   and J :: "qtname"   and G :: "prog"   and jm :: "qtname \<times> mhead"   and sig :: "sig" assumes "G\<turnstile>I\<preceq>I J"   and "wf_prog G"   and "is_iface G J"   and "jm \<in> DeclConcepts.imethds G J sig" shows "\<exists>im\<in>DeclConcepts.imethds G I sig. is_static im = is_static jm \<and> accmodi im = accmodi jm \<and> G\<turnstile>resTy im\<preceq>resTy jm"
simps_18: fixes vb :: "'a"   and vc :: "value list"   and v :: "bool" shows "local.less_gexp_aux (In vb vc) (Bc v) = False"
minGraphProps10: fixes g :: "graph"   and a :: "nat"   and b :: "nat" assumes "minGraphProps g"   and "(a, b) \<in> \<E> g" shows "(b, a) \<in> \<E> g"
list_eq_prod: fixes m :: "real"   and xs :: "real list" shows "prod_list (list_eq xs m) = m ^ length (list_eq xs m)"
induct: fixes P :: "(fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option) \<Rightarrow> (key \<times> fresh_t) signal \<Rightarrow> bool"   and a0 :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and a1 :: "(key \<times> fresh_t) signal" assumes "\<And>runz B A Kab Nb. P runz (Commit [B, A] (Kab, Nb))"   and "\<And>runz B A Kab Nb. P runz (Running [B, A] (Kab, Nb))"   and "\<And>runz va. P runz (Running [] va)"   and "\<And>runz vb va. P runz (Running [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Running (vb # vd # vf # vg) va)"   and "\<And>runz va. P runz (Commit [] va)"   and "\<And>runz vb va. P runz (Commit [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Commit (vb # vd # vf # vg) va)" shows "P a0 a1"
full_sel: fixes K :: "'a set"   and \<phi> :: "'a" shows "\<gamma>\<^sub>F\<^sub>C K \<phi> \<equiv> if K .\<bottom>. \<phi> = {} then {K} else K .\<bottom>. \<phi>"
ccField_CCfix: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccField (CCfix \<Gamma>\<cdot>(ae, G)) \<subseteq> fv \<Gamma> \<union> ccField G"
iso_same_card: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "G \<cong> H" shows "card (carrier G) = card (carrier H)"
blinfun_compose_zero_2: fixes x :: "'d \<Rightarrow>\<^sub>L 'c" shows "x o\<^sub>L 0 = 0"
rel_fun_pos_distr_iff: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and tytok :: "('h \<times> 'i \<times> 'j) itself" shows "rel_fun_pos_distr_cond A A' tytok = True"
wod_backward_slice_valid_node: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and n :: "'g"   and S :: "'g set" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val"   and "n \<in> CFG_wf.wod_backward_slice sourcenode targetnode valid_edge Def Use S" shows "CFG.valid_node sourcenode targetnode valid_edge n"
connex_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "connex B r"
augmenting_path_contains_resCap: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" obtains e :: "nat \<times> nat" where "e \<in> set p" and "cf e = resCap p"
select_convs_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
subcls1I: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and rest :: "fdecl list \<times> 'a mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
assms_1: shows "FO \<subseteq> {0..<n}"
no_\<tau>move1_\<tau>s_to_no_\<tau>move2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "\<And>s1'. \<not> \<tau>mred1 s1 s1'" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> (\<forall>s2''. \<not> \<tau>mred2 s2' s2'') \<and> s1 \<approx>m s2'"
\<pi>\<pi>'_2: shows "wfp UNIV \<pi>'"
undef_def: shows "undef \<equiv> (x, y) + r *\<^sub>R (i, 0::'b)"
abs_eq: fixes x :: "int" shows "Uint32 (integer_of_int x) = Abs_uint32 (word_of_int x)"
iT_finite_1: fixes n :: "nat" shows "finite [\<dots>n]"
strict_order_iff: fixes a :: "'g"   and b :: "'g" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
poly_roots_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "poly_roots (p * q) = poly_roots p + poly_roots q"
assms_2: shows "s1 \<in> d_reachable M (initial M)"
bind_fv2_renamed_var: fixes y :: "variable"   and t :: "Core.term"   and \<tau> :: "typ"   and i :: "nat"   and x :: "variable" assumes "y \<notin> fst ` fv t" shows "bind_fv2 (y, \<tau>) i (subst_term [((x, \<tau>), Fv y \<tau>)] t) = bind_fv2 (x, \<tau>) i t"
rcset: fixes x :: "'c cset" shows "rcset x \<in> {A. countable A}"
butterfly_def: fixes s :: "'e"   and t :: "'d" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
mhd_simp: fixes r :: "ref_ty"   and m :: "mhead" shows "mhd (r, m) = m"
image_Union: fixes f :: "'e \<Rightarrow> 'd"   and S :: "'e set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
dependent_eq_real: fixes S :: "'c set" assumes "real_vector_space_on S" shows "module_on.dependent (*\<^sub>R) = dependent"
nF_def: fixes s :: "'a" shows "nF s \<equiv> card {f \<in> F. face f s \<and> compo' f}"
det_lowerdiagonal: fixes A :: "(('d, 'c) vec, 'c) vec" assumes "\<And>i j. i < j \<Longrightarrow> A $ i $ j = (0::'d)" shows "Determinants.det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
binomial_ring: fixes a :: "'c"   and b :: "'c"   and n :: "nat" shows "(a + b) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * a ^ k * b ^ (n - k))"
univ_poly_is_euclidean: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "euclidean_domain (K [X]\<^bsub>K [X]\<^esub>) degree"
list_emb_trans: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "list_emb P xs ys"   and "list_emb P ys zs" shows "list_emb P xs zs"
assms_3: shows "v \<prec>\<^sub>t lt p"
openin_contains_cball: fixes T :: "'a set"   and S :: "'a set" shows "openin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x\<in>S. \<exists>e>0. cball x e \<inter> T \<subseteq> S))"
tab_simps_4: shows "local.dom \<tau>\<mu>.tab = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1"
finite_Fvars_fm: fixes A :: "fm" shows "finite (Fvars A)"
matched_bracket_param_6: shows "n\<^sub>3 -p:V'\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4"
lunit'_in_vhom: fixes f :: "'a"   and b :: "'a" assumes "B.ide f"   and "trg\<^sub>B f = b" shows "\<guillemotleft>B.lunit' f : f \<rightarrow>\<^sub>B b \<star>\<^sub>B f\<guillemotright>"
AAss_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e'' :: "'addr expr1"   and v :: "'addr val" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<lfloor>e'\<rceil> := e'') h (stk @ [v], loc, length (compE2 e) + pc, xcp) (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
all_forward: fixes P' :: "'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<forall>x. P x"
Cod_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "Cod (Dom t) = Dom t"
bounded_linear_inner_left: fixes b :: "'a" shows "bounded_linear (\<lambda>a. inner a b)"
greatest_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_fixpoint f" shows "is_greatest_fixpoint f (\<nu> f)"
I: shows "\<nexists>b. Pri_AgrK b \<in> A \<and> ExtMapK (S (User m, n, run)) = Some b"
simps: fixes uu :: "'a tree"   and r :: "'a"   and uv :: "'a tree" shows "root \<langle>uu, r, uv\<rangle> = r"
nneg_def: fixes P :: "'c \<Rightarrow> 'd" shows "nneg P = (\<forall>x. (0::'d) \<le> P x)"
encode_problem_parallel_complete_vi_a: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and k :: "nat"   and op :: "'b strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1" shows "valuation_for_plan \<Pi> \<pi> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"
FMIQ: shows "findMin q \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q))"
execs_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c # cs,css,s\<rangle> \<Rightarrow> t" obtains ta :: "('e, 'f) xstate" where "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> ta" and "\<Gamma>\<turnstile> \<langle>cs,css,ta\<rangle> \<Rightarrow> t"
assms_4: fixes T :: "'a set"   and s :: "'b"   and x :: "'a" assumes "T \<subseteq> S"   and "I T s"   and "x \<in> S"   and "x \<notin> T" shows "f x s \<le> SPEC (I (insert x T))"
simps_16: fixes b :: "'c exp"   and e :: "'c exp" shows "ss_exp (while (b) e) = while (b) e"
tuple_ext: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<not> span f g" shows "HF'.tuple f g = null"
power2_eq_square: fixes a :: "'b" shows "nataction.power2 a = a + a"
nprv_prvI: fixes \<phi> :: "'fmla" assumes "nprv {} \<phi>"   and "\<phi> \<in> fmla" shows "prv \<phi>"
map_gpv_conv_map_gpv': fixes f :: "'g \<Rightarrow> 'j"   and g :: "'h \<Rightarrow> 'k" shows "map_gpv f g = map_gpv' f g id"
cond_symm: fixes b :: "'b upred"   and P :: "('a, 'b) uexpr"   and Q :: "('a, 'b) uexpr" shows "P \<triangleleft> b \<triangleright> Q = Q \<triangleleft> \<not> b \<triangleright> P"
separatedin_subtopology: fixes X :: "'b topology"   and U :: "'b set"   and S :: "'b set"   and T :: "'b set" shows "separatedin (subtopology X U) S T = (S \<subseteq> U \<and> T \<subseteq> U \<and> separatedin X S T)"
long_divides_1: shows "p = [\<one>, \<ominus> x] \<otimes>\<^bsub>poly_ring R\<^esub> q \<oplus>\<^bsub>poly_ring R\<^esub> r"
confTs_Cons1: fixes P :: "'a prog"   and h :: "heap"   and x :: "val"   and xs :: "val list"   and ys :: "ty err list" shows "(P,h \<turnstile> x # xs [:\<le>\<^sub>\<top>] ys) = (\<exists>z zs. ys = z # zs \<and> P,h \<turnstile> x :\<le>\<^sub>\<top> z \<and> P,h \<turnstile> xs [:\<le>\<^sub>\<top>] zs)"
inverse_less_iff_less: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "(inverse a < inverse b) = (b < a)"
Span_closed: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set ms \<subseteq> V" shows "R_scalar_mult.RSpan UNIV smult ms \<subseteq> V"
assms: shows "interaction_bounded_by' (\<lambda>_. True) gpv n"
integrable_def: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "integrable f M = (measure_space M \<and> (\<exists>x. x \<in> nnfis (pp f) M) \<and> (\<exists>y. y \<in> nnfis (np f) M))"
PiE_mono: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and C :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
Tree_wf_eqvt_aux: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree"   and p :: "perm" assumes "(t1, t2) \<in> Tree_wf" shows "(p \<bullet> t1, p \<bullet> t2) \<in> Tree_wf"
lim: fixes x :: "'a" assumes "x \<in> S - N" shows "(\<lambda>n. f n x) \<longlonglongrightarrow> g x"
res_pow_eq: fixes x :: "int"   and n :: "nat" shows "x [^] n = x ^ n mod m"
qmsg_wf: shows "wellformed \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
suf_exp_le: fixes u :: "'b list"   and m :: "nat"   and n :: "nat" assumes "u \<noteq> \<epsilon>"   and "u \<^sup>@ m \<le>s u \<^sup>@ n" shows "m \<le> n"
rbt_product_def: fixes f :: "'f \<Rightarrow> 'i \<Rightarrow> 'g \<Rightarrow> 'j \<Rightarrow> 'h"   and rbt1 :: "('f, 'i) rbt"   and rbt2 :: "('g, 'j) rbt" shows "rbt_product f rbt1 rbt2 = rbtreeify (alist_product f (RBT_Impl.entries rbt1) (RBT_Impl.entries rbt2))"
u1u2_3: shows "t2 \<in> {t. linepath a b t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
one_chain1_ps_valid: shows "\<forall>(k, \<gamma>)\<in>one_chain1 - ps1. valid_path \<gamma>"
Sup_upper: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and x :: "'b::type"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(x::'b::type) \<in> (A::'b::type set)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((Sup::'b::type set \<Rightarrow> 'b::type) (A::'b::type set))"
bij_betw_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
floor_less_iff: fixes x :: "'c"   and z :: "int" shows "(\<lfloor>x\<rfloor> < z) = (x < of_int z)"
encRelS: shows "\<forall>S1 S2. \<lbrakk>S1\<rbrakk> = \<lbrakk>S2\<rbrakk> \<longrightarrow> (S1, S2) \<in> RelS\<^sup>+"
elementary_interval: fixes a :: "'a"   and b :: "'a" obtains p :: "'a set set" where "p division_of cbox a b"
Start_nsys_xcpts1_3: shows "Start \<noteq> OutOfMemory"
vector_scalar_commute: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and c :: "'a"   and x :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v (c *s x) = c *s (A *v x)"
biglemma: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and h :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "n \<noteq> 0"   and "compactin (Euclidean_space n) S"   and "continuous_map (subtopology (Euclidean_space n) S) (Euclidean_space n) h"   and "inj_on h S" shows "path_connectedin (Euclidean_space n) (topspace (Euclidean_space n) - h ` S) = path_connectedin (Euclidean_space n) (topspace (Euclidean_space n) - S)"
hermite_cnj_of_tensor: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" shows "A \<Otimes> B\<^sup>\<dagger> = A\<^sup>\<dagger> \<Otimes> B\<^sup>\<dagger>"
step_sim: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "(a, a') \<in> s1.a.E"   and "(a, b) \<in> R1" shows "\<exists>b'. (b, b') \<in> s1.b.E \<and> (a', b') \<in> R1"
hom: shows "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
has_density_embed_measure'': fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and f' :: "'d \<Rightarrow> 'c"   and M :: "'d measure"   and \<delta> :: "'d \<Rightarrow> ennreal" assumes "inj f"   and "\<And>x. x \<in> space N \<Longrightarrow> f' (f x) = x"   and "has_density M (embed_measure N f) \<delta>" shows "has_density (distr M N f') N (\<delta> \<circ> f)"