hcju/isabelleps huggingface dataset:
encF: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap"   and "f \<in> F" shows "encF f \<in> Ops"
eq_reflection: fixes x :: "'c"   and y :: "'c" assumes "x = y" shows "x \<equiv> y"
kbo_std_basis_axioms_def: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and wt_sym :: "'a \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'a \<Rightarrow> (('a, 'b) tm \<Rightarrow> ('a, 'b) tm \<Rightarrow> bool) \<Rightarrow> ('a, 'b) tm list \<Rightarrow> ('a, 'b) tm list \<Rightarrow> bool" shows "kbo_std_basis_axioms gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf \<equiv> ((0 < \<epsilon> \<and> \<delta> \<le> \<epsilon>) \<and> (\<forall>\<zeta>. 0 < \<delta> \<longrightarrow> arity.arity_hd arity_sym arity_var \<zeta> \<noteq> \<infinity>) \<and> (\<forall>f. of_nat (\<epsilon> - the_enat (of_nat \<delta> * arity_sym f)) \<le> wt_sym f) \<and> (\<forall>f g. arity_sym f = 1 \<longrightarrow> wt_sym f = (0::'c) \<longrightarrow> gt_sym f g \<or> g = f)) \<and> ((\<forall>f. arity_sym f = 1 \<longrightarrow> wt_sym f = (0::'c) \<longrightarrow> \<delta> = \<epsilon>) \<and> (\<forall>f. ext_irrefl_before_trans (extf f))) \<and> (\<forall>f. ext_compat_list_strong (extf f)) \<and> (\<forall>f. ext_hd_or_tl (extf f)) \<and> (\<forall>f. \<delta> = \<epsilon> \<longrightarrow> ext_snoc (extf f))"
code_dec_morph: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> \<langle>\<C>\<rangle>"   and "y \<in> \<langle>\<C>\<rangle>" shows "(Dec \<C> x) \<cdot> (Dec \<C> y) = Dec \<C> x \<cdot> y"
mat_t_mat_t: fixes A :: "'a \<times> 'b \<times> 'c \<times> 'd" shows "mat_transpose (mat_transpose A) = A"
is_process_REP_D: fixes P :: "'c process"   and Q :: "'c process" shows "is_process ({(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}, D P \<union> D Q)"
simps_2: fixes s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" shows "vars_mset (App s t) = vars_mset s + vars_mset t"
simps_5: fixes a :: "Cooper.num"   and b :: "Cooper.num" shows "zsplit0 (Add a b) = (let (ia, a') = zsplit0 a; (ib, b') = zsplit0 b in (ia + ib, Add a' b'))"
p1_2: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> p1" shows "x \<in> K"
set_uprod_simps: fixes x :: "'b"   and y :: "'b" shows "set_uprod (Upair x y) = {x, y}"
mod_0: fixes a :: "'b" shows "(0::'b) mod a = (0::'b)"
divide_ennreal: fixes r :: "real"   and q :: "real" assumes "0 \<le> r"   and "0 < q" shows "ennreal r / ennreal q = ennreal (r / q)"
ode_alt_sem: fixes ODE :: "('a, 'b) ODE"   and \<phi> :: "('a, 'c, 'b) formula"   and I :: "('a, 'c, 'b) interp" assumes "osafe ODE"   and "fsafe \<phi>" shows "prog_sem I (EvolveODE ODE \<phi>) = {(\<nu>, mk_v I ODE \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE \<nu> x \<in> fml_sem I \<phi>} \<and> VSagree (sol 0) (fst \<nu>) {x |x. Inl x \<in> FVP (EvolveODE ODE \<phi>)}}"
enforce_map_spmf: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "enforce_spmf P (map_spmf f p) = map_spmf f (enforce_spmf (P \<circ> f) p)"
post_in_annos: fixes C :: "'a acom" shows "post C \<in> set (annos C)"
InputS_def: fixes x1 :: "name" shows "InputS x1 \<equiv> Abs_subject (InputS_Rep x1)"
swap_msgs_Recv_Send: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isSend ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
assu3: shows "LIM n sequentially. d n ^ 2 ^ n / real_of_int (b n) :> at_top"
similar'_botI: fixes n :: "nat" shows "\<bottom> \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<bottom>"
inf_bot_right: fixes x :: "'a" shows "x \<sqinter> \<bottom> = \<bottom>"
ldistinct_LCons: fixes x :: "'c"   and xs :: "'c llist" shows "ldistinct (LCons x xs) = (x \<notin> lset xs \<and> ldistinct xs)"
pderiv_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "pderiv (p * q) = p * pderiv q + q * pderiv p"
embed_bool_bounded: fixes P :: "'b \<Rightarrow> bool" shows "Expectations.bounded \<guillemotleft> P \<guillemotright>"
img_mkPoint_1: fixes a :: "'s" assumes "ide a" shows "mkPoint a \<in> local.set a \<rightarrow> hom unity a"
remove: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
chine_assoc'_in_hom: shows "\<guillemotleft>fgh.chine_assoc' : HfHgh.chine \<rightarrow>\<^sub>C HHfgh.chine\<guillemotright>"
last_seg_impl_def_opt: fixes s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "last_seg_impl s \<equiv> case s of (S, B, I, P) \<Rightarrow> ASSERT (length B - 1 < length B) \<bind> (\<lambda>_. WHILE\<^sub>T (\<lambda>(l, uu_). l < (if length B - 1 + 1 = length B then length S else B ! (length B - 1 + 1))) (\<lambda>(l, res). ASSERT (l < length S) \<bind> (\<lambda>_. let x = S ! l in ASSERT (x \<notin> res) \<bind> (\<lambda>_. RETURN (Suc l, insert x res)))) (B ! (length B - 1), {}) \<bind> (\<lambda>(uu_, res). RETURN res))"
Ex_last_occurrence: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat"   and l :: "nat" assumes "P n"   and "n \<le> l" shows "\<exists>m. P (l - m) \<and> (\<forall>i<m. \<not> P (l - i))"
irreducible\<^sub>dD_2: fixes p :: "'c poly"   and q :: "'c poly"   and r :: "'c poly" assumes "irreducible\<^sub>d p"   and "degree q < degree p"   and "degree r < degree p" shows "p \<noteq> q * r"
blinding_of_on_prod: fixes A :: "'c set"   and rha :: "'c \<Rightarrow> 'd"   and boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'e set"   and rhb :: "'e \<Rightarrow> 'f"   and bob :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "blinding_of_on A rha boa"   and "blinding_of_on B rhb bob" shows "blinding_of_on {x. fsts x \<subseteq> A \<and> snds x \<subseteq> B} (map_prod rha rhb) (rel_prod boa bob)"
that: fixes r :: "'val"   and s :: "'val" assumes "phiArg g p r"   and "phiArg g p s"   and "distinct [p, r, s]" shows "thesis"
1: shows "\<forall>j\<in>local.underS i. f j = g j"
nn_integral_add: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
dim_sums_Int: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "vec.subspace S"   and "vec.subspace T" shows "vec.dim {x + y |x y. x \<in> S \<and> y \<in> T} + vec.dim (S \<inter> T) = vec.dim S + vec.dim T"
list_relE2: fixes l :: "'b list"   and R :: "('a \<times> 'b) set" assumes "([], l) \<in> \<langle>R\<rangle>list_rel" obtains "l = []"
authTicket_form: fixes A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "authK \<notin> range shrK \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace>"
ideD_2: fixes a :: "'a" assumes "ide a" shows "local.dom a = a"
r_and: fixes x :: "nat"   and y :: "nat" shows "eval r_and [x, y] \<down>= (if x = 0 \<and> y = 0 then 0 else 1)"
assms_2: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "P t\<^sub>\<alpha>" shows "P (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
upper_triangular_mat_delete_make_mat2: fixes k :: "nat"   and n :: "nat"   and B :: "'a mat" assumes "k < n" shows "Matrix.upper_triangular (mat_delete (mat_delete (make_mat n k B) 0 k) (k - 1) 0)"
CFG_SSA_wf_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option" shows "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_wf_axioms \<alpha>n invar inEdges' Entry defs uses phis"
inc_not_0: fixes n :: "Arity" shows "(inc\<cdot>n = 0) = False"
simps: fixes a1 :: "'a multiset"   and a2 :: "'a multiset"   and ns :: "('a \<times> 'a) set" shows "((a1, a2) \<in> multpw ns) = (a1 = {#} \<and> a2 = {#} \<or> (\<exists>x y X Y. a1 = add_mset x X \<and> a2 = add_mset y Y \<and> (x, y) \<in> ns \<and> (X, Y) \<in> multpw ns))"
HomFtorOpObj: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "(Hom\<^bsub>C\<^esub>[\<midarrow>,X]) @@ Y = Hom\<^bsub>C\<^esub> Y X"
true_cls_mono: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and I :: "'a set" assumes "set_mset C \<subseteq> set_mset D"   and "I \<Turnstile> C" shows "I \<Turnstile> D"
Language_delta: fixes s :: "'s"   and a :: "'a" assumes "wellformed s"   and "a \<in> set alphabet" shows "Language (delta a s) = \<dd> (Language s) a"
add_edge_def: fixes v :: "'d"   and e :: "'e"   and v' :: "'d"   and g :: "('d, 'e, 'f) graph_scheme" shows "add_edge v e v' g \<equiv> \<lparr>nodes = {v, v'} \<union> nodes g, edges = insert (v, e, v') (edges g)\<rparr>"
sconsts_sabs: fixes cs :: "(Term_Class.term \<times> sterm) list"   and S :: "name fset" shows "(consts (Sabs cs) |\<subseteq>| S) = list_all (\<lambda>(uu_, t). consts t |\<subseteq>| S) cs"
c_def: shows "c = inverse (2 * y ^ (2 * n + 1))"
diagseq_seqseq: fixes k :: "nat" shows "diagseq \<circ> (+) k = seqseq k \<circ> (\<lambda>x. fold_reduce k x (k + x))"
select_convs_1: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprFst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprFst"
less: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s"   and "b \<in> s"   and "a i < b i" shows "a < b"
rGamma_eq_zero_iff: fixes z :: "'a" shows "(rGamma z = (0::'a)) = (z \<in> \<int>\<^sub>\<le>\<^sub>0)"
listIntersectionWithSet: fixes l :: "'a list"   and X :: "'a set" shows "set (takeAll (\<lambda>x. x \<in> X) l) \<subseteq> X \<inter> set l"
rquot_mult: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "partial_semigroup.gR_rel times D x y"   and "z = cancellative_partial_semigroup.rquot times D y x" shows "times x z = y"
prop_atoms_entailment_iff: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<A> :: "'b ltln set" assumes "\<phi> \<in> prop_atoms \<psi>" shows "\<A> \<Turnstile>\<^sub>P \<phi> = (\<phi> \<in> \<A>)"
coprime_common_divisor: 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 b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one a b"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "dvd.dvd times c one"
vg_int: shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
cut_le_not_empty_imp: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>\<le> t \<noteq> {}" shows "I \<noteq> {}"
lm022: fixes X :: "'c set"   and mx :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite X"   and "mx \<in> X"   and "\<forall>x\<in>X - {mx}. f x < f mx" shows "argmax f X = {mx}"
simps_3: fixes x1 :: "'b" shows "tick \<noteq> ev x1"
eq_rep_abs: fixes x :: "cmsg" shows "eq x (Re (Ab x))"
that: fixes u :: "'c \<Rightarrow> real" assumes "s = (l, u)"   and "u \<in> [u]\<^sub>\<R>"   and "[u]\<^sub>\<R> \<in> \<R>"   and "R = [u]\<^sub>\<R>" shows "thesis"
denote_measure_positive_trace_dim: fixes dims :: "nat list"   and x1 :: "nat"   and x2a :: "nat \<Rightarrow> complex mat"   and x3a :: "com list"   and x4 :: "complex mat" assumes "state_sig.well_com dims (Measure x1 x2a x3a)"   and "x4 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator x4"   and "\<And>x3aa \<rho>. \<lbrakk>x3aa \<in> set x3a; state_sig.well_com dims x3aa; \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims); partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (state_sig.denote dims x3aa \<rho>) \<and> trace (state_sig.denote dims x3aa \<rho>) \<le> trace \<rho> \<and> state_sig.denote dims x3aa \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)" shows "positive (state_sig.denote dims (Measure x1 x2a x3a) x4) \<and> trace (state_sig.denote dims (Measure x1 x2a x3a) x4) \<le> trace x4 \<and> state_sig.denote dims (Measure x1 x2a x3a) x4 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
that: fixes T :: "'b set" assumes "closed T"   and "S homeomorphic T" shows "thesis"
sumhr_const: fixes n :: "nat star"   and r :: "real" shows "sumhr (0, n, \<lambda>i. r) = hypreal_of_hypnat n * hypreal_of_real r"
causal_delivery: fixes m2 :: "'a \<times> ('a, 'b) operation"   and j :: "nat"   and m1 :: "'a \<times> ('a, 'b) operation" assumes "Deliver m2 \<in> set (history j)"   and "hb m1 m2" shows "Deliver m1 \<sqsubset>\<^sup>j Deliver m2"
extremum_unique: fixes a :: "'d" shows "(a \<le> \<bottom>) = (a = \<bottom>)"
cont_ltl: shows "cont lSup (\<sqsubseteq>) lSup (\<sqsubseteq>) ltl"
finfun_update_const_same: fixes b :: "'b"   and a :: "'a" shows "finfun_update (finfun_const b) a b = finfun_const b"
set_split_bst1: fixes t :: "'a tree"   and x :: "'a" assumes "bst t" shows "set_tree (fst (split_bst x t)) = {y \<in> set_tree t. y < x}"
simps_4: shows "dist_param_type UniformInt = PRODUCT INTEG INTEG"
vsv_vimageI2': fixes r :: "V"   and b :: "V"   and a :: "V" assumes "vsv r"   and "b = r\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
runit'_in_vhom: fixes f :: "'a"   and a :: "'a" assumes "B.ide f"   and "src\<^sub>B f = a" shows "\<guillemotleft>B.runit' f : f \<Rightarrow>\<^sub>B f \<star>\<^sub>B a\<guillemotright>"
subcomplex_pgallery: fixes X :: "'b set set"   and Y :: "'b set set"   and Cs :: "'b set list" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y"   and "SimplicialComplex.pmaxsimpchain Y Cs" shows "SimplicialComplex.pmaxsimpchain X Cs"
residual_network_sel_4: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_network \<Delta>" shows "sink (countable_network.residual_network \<Delta> f) = sink \<Delta>"
map_scale_assoc: fixes c :: "'c"   and d :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> d \<cdot> t = (c * d) \<cdot> t"
mimg_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('h, 'f, 'i) Module_scheme"   and N :: "('e, 'f, 'j) Module_scheme"   and f :: "'h \<Rightarrow> 'e" shows "mimg\<^bsub>R M,N\<^esub> f = mdl N (f ` carrier M)"
bij_is_maprop: fixes x :: "'a" assumes "is_bij x"   and "is_map x" shows "x\<^sup>\<smile> ; x = 1' \<and> x ; x\<^sup>\<smile> = 1'"
uses'_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and n :: "'node" shows "uninst.uses' u p g n \<equiv> uninst.substNext u p g ` u g n"
not_integrable_integral_eq: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<not> integrable M f" shows "integral\<^sup>L M f = (0::'d)"
mset_heap_Cons: fixes t :: "'b tree"   and ts :: "'b tree list" shows "mset_heap (t # ts) = mset_tree t + mset_heap ts"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
gdelta_imp_fsigma: fixes S :: "'b set" assumes "gdelta S" shows "fsigma (- S)"
span_diff: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x - y \<in> ideal S"
ag_r_zero: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = x"
intensional_rews_5: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and w :: "'d" shows "(\<forall>x. A x) w \<equiv> \<forall>x. A x w"
refl: fixes x :: "'a" assumes "x \<in> carrier L" shows "x .=\<^bsub>L\<^esub> x"
opt_GSdef: fixes f :: "'b \<times> 'c \<times> 'd \<times> 'e \<Rightarrow> 'f"   and g :: "'b \<times> 'c \<times> 'd \<times> 'e \<Rightarrow> 'f"   and s :: "'b \<times> 'c \<times> 'd \<times> 'e" assumes "f \<equiv> g" shows "f s \<equiv> case s of (S, B, I, P) \<Rightarrow> g (S, B, I, P)"
less_top: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and top :: "'k"   and a :: "'k" assumes "class.order_top less_eq less top" shows "(a \<noteq> top) = less a top"
Bernstein_sum: fixes c :: "real"   and d :: "real"   and P :: "real poly"   and p :: "nat" assumes "c \<noteq> d"   and "degree P \<le> p" shows "P = (\<Sum>j = 0..p. smult (inverse (real (p choose j)) * coeff (reciprocal_poly p (P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:]) \<circ>\<^sub>p [:1, 1:]) (p - j)) (Bernstein_Poly j p c d))"
right_unique_cr_spmf_gpv: shows "right_unique cr_spmf_gpv"
order_less_le: fixes x :: "'d"   and y :: "'d" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
not_eq_order_implies_strict: fixes a :: "'t"   and b :: "'t" assumes "a \<noteq> b"   and "le_of_nat_term_order cmp_term b a" shows "lt_of_nat_term_order cmp_term b a"
lookup_tensor_from_lookup: fixes "is" :: "nat list"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'm" assumes "is \<lhd> ds" shows "lookup (tensor_from_lookup ds e) is = e is"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and N :: "'b ltln set" shows "\<phi> and\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2 and\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2"
BW_def: shows "BW = (\<lambda>n. first_entrance_set A n \<inter> (T ^^ n) --` W)"
valid_tmap_def: fixes t :: "(int \<times> 'b) Tree" shows "valid_tmap t \<equiv> sortedTree fst t"
g: fixes x :: "real" assumes "0 < x"   and "x < a" shows "(g has_real_derivative g' x) (at x)"
wtE_not_glitOfL: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and l :: "('fsym, 'psym) lit" assumes "Ik.wtE \<xi>" shows "\<not> GE.satC \<xi> (list (glitOfL l))"
real_cond_exp_charact: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "\<And>A. A \<in> sets F \<Longrightarrow> set_lebesgue_integral M A f = set_lebesgue_integral M A g"   and "integrable M f"   and "integrable M g"   and "g \<in> borel_measurable F" shows "AEeq M (real_cond_exp M F f) g"
morE1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "('f, 'g, 'h) F1 \<Rightarrow> 'g"   and s2' :: "('f, 'g, 'h) F2 \<Rightarrow> 'h"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and z :: "('f, 'd, 'e) F1" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "z \<in> F1in UNIV B1 B2" shows "f (s1 z) = s1' (F1map id f g z)"
Sigma_def: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" shows "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {(x, y)}"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "S.UP f = S.UP f'" shows "f = f'"
rel_option_None1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option" shows "rel_option P None x = (x = None)"
snth_smap: fixes f :: "'d \<Rightarrow> 'c"   and s :: "'d stream"   and n :: "nat" shows "smap f s !! n = f (s !! n)"
C1_sub_C: shows "CT \<turnstile> C1 <: C"
wary_fo_args: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat" assumes "s \<in> set (args t)"   and "arity.wary_fo arity_sym arity_var t" shows "arity.wary_fo arity_sym arity_var s"
assms: shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> Undecided"
elims: fixes x :: "('c \<Rightarrow> 'query \<Rightarrow> ('response \<times> 'c) spmf) \<times> 'c"   and xa :: "('pubkey \<Rightarrow> (('plain \<times> 'plain) \<times> 'd, 'query, 'response) gpv) \<times> ('cipher \<Rightarrow> 'd \<Rightarrow> (bool, 'query, 'response) gpv)"   and y :: "real" assumes "advantage x xa = y" obtains "oracle" :: "'c \<Rightarrow> 'query \<Rightarrow> ('response \<times> 'c) spmf" and \<sigma> :: "'c" and \<A> :: "('pubkey \<Rightarrow> (('plain \<times> 'plain) \<times> 'd, 'query, 'response) gpv) \<times> ('cipher \<Rightarrow> 'd \<Rightarrow> (bool, 'query, 'response) gpv)" where "x = (oracle, \<sigma>)" and "xa = \<A>" and "y = \<bar>spmf (run_gpv oracle (game \<A>) \<sigma>) True - 1 / 2\<bar>"
d_gchain_mem_sg: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "d_gchain G n f" shows "\<forall>i\<le>n. G \<guillemotright> f i "
isolated_singularity_at_holomorphic: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and z :: "complex" assumes "f holomorphic_on s - {z}"   and "open s"   and "z \<in> s" shows "isolated_singularity_at f z"
l1: shows "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
swap_subst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y" shows "X #[Y / y]_ys #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
insert_absorb: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "Set.insert a A = A"