hcju/isabelleps huggingface dataset:
eo: fixes i :: "nat" assumes "i < n" shows "exponential_order M c (nderiv i f {0<..})"
FVar_lemma1: fixes G :: "prog"   and statC :: "qtname"   and fn :: "vname"   and statDeclC :: "qtname"   and f :: "Decl.field"   and x2 :: "abopt"   and s2 :: "st"   and a :: "val"   and y :: "class"   and L :: "lenv"   and s1 :: "st" assumes "table_of (DeclConcepts.fields G statC) (fn, statDeclC) = Some f"   and "x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq>Class statC"   and "wf_prog G"   and "G\<turnstile>statC\<preceq>\<^sub>C statDeclC"   and "statDeclC \<noteq> Object"   and "class G statDeclC = Some y"   and "(x2, s2)\<Colon>\<preceq>(G, L)"   and "s1\<le>|s2"   and "inited statDeclC (globs s1)"   and "(if static f then id else np a) x2 = None" shows "\<exists>obj. globs s2 (if static f then Inr statDeclC else Inl (the_Addr a)) = Some obj \<and> var_tys G (tag obj) (if static f then Inr statDeclC else Inl (the_Addr a)) (Inl (fn, statDeclC)) = Some (type f)"
alice_meas_def: fixes \<phi> :: "complex Matrix.mat" shows "alice_meas \<phi> = [(prob0 3 (alice \<phi>) 0 * prob0 3 (post_meas0 3 (alice \<phi>) 0) 1, post_meas0 3 (post_meas0 3 (alice \<phi>) 0) 1), (prob0 3 (alice \<phi>) 0 * prob1 3 (post_meas0 3 (alice \<phi>) 0) 1, post_meas1 3 (post_meas0 3 (alice \<phi>) 0) 1), (prob1 3 (alice \<phi>) 0 * prob0 3 (post_meas1 3 (alice \<phi>) 0) 1, post_meas0 3 (post_meas1 3 (alice \<phi>) 0) 1), (prob1 3 (alice \<phi>) 0 * prob1 3 (post_meas1 3 (alice \<phi>) 0) 1, post_meas1 3 (post_meas1 3 (alice \<phi>) 0) 1)]"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := P\<rparr>)" shows "H \<subseteq> carrier (G\<lparr>carrier := P\<rparr>)"
similar'_down_le: fixes n :: "nat"   and m :: "nat"   and d :: "Value"   and e :: "CValue" assumes "n \<le> m"   and "\<psi>\<^sup>D\<^bsub>m\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>m\<^esub> \<psi>\<^sup>E\<^bsub>m\<^esub>\<cdot>e" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e"
preserves_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" shows "F (g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f) = F g \<cdot>\<^sub>B F f"
BWN: shows "bounded_WHILE n cond step m0 = RES Xh"
of_name_inj: fixes of_name :: "name \<Rightarrow> 'c"   and name\<^sub>1 :: "name"   and name\<^sub>2 :: "name" assumes "class.is_name of_name"   and "of_name name\<^sub>1 = of_name name\<^sub>2" shows "name\<^sub>1 = name\<^sub>2"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>xs. P 0 xs"   and "\<And>n x xs. \<lbrakk>\<And>xa xb y. \<lbrakk>xa = heapify (Suc n div 2) xs; (xb, y) = xa\<rbrakk> \<Longrightarrow> P (Suc n div 2) xs; \<And>xa xb y xaa xab xac ya. \<lbrakk>xa = heapify (Suc n div 2) xs; (xb, y) = xa; xaa = t_heapify (Suc n div 2) xs; xab = heapify (n div 2) y; (xac, ya) = xab\<rbrakk> \<Longrightarrow> P (n div 2) y\<rbrakk> \<Longrightarrow> P (Suc n) (x # xs)"   and "\<And>v. P (Suc v) []" shows "P a0 a1"
cl11_d: fixes x :: "'a" shows "d (x \<sqinter> nc) \<cdot> nc = (x \<sqinter> nc) \<cdot> nc"
opt_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "opt A \<subseteq> opt B"
run_asset_ii_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_asset_ii n s \<equiv> SOME x. pred_asset_ii n (run_owner_ii n s) x"
indoc_appendE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "traces.indoc ind a (u @ v)" obtains "a \<in> set u" and "traces.indoc ind a u"   | "a \<notin> set u" and "\<forall>a\<in>{a}. \<forall>b\<in>set u. ind a b" and "traces.indoc ind a v"
dterms_no_call: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "call(pn) \<notin> dterms \<Gamma> p"
continuous_on_arcsin': shows "continuous_on {- 1..1} arcsin"
Path_not_Null_iff: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and q :: "ref"   and as :: "ref list" assumes "p \<noteq> Null" shows "Path p h q as = (as = [] \<and> q = p \<or> (\<exists>ps. as = p # ps \<and> Path (h p) h q ps))"
intFF_def: shows "intFF \<equiv> InfModel.intFF TE_arOf TE_resOf intTI intFI"
factorial_monoid_axioms: shows "factorial_monoid (Ring_Divisibility.mult_of R)"
powsum_mpoly_recurrence: fixes k :: "nat"   and X :: "nat set" assumes "0 < k"   and "finite X" shows "powsum_mpoly X k = (- 1) ^ (k + 1) * of_nat k * sym_mpoly X k - (\<Sum>i = 1..<k. (- 1) ^ i * sym_mpoly X i * powsum_mpoly X (k - i))"
arr_dom: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.arr (Cop.dom f)"
hoaret_augment_context: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and p :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and \<Theta>' :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P p Q,A"   and "\<Theta> \<subseteq> \<Theta>'" shows "\<Gamma>,\<Theta>'\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P p Q,A"
skip: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d" shows "\<Gamma>,\<gamma>,p\<turnstile> [] \<Rightarrow>\<^sub>r Undecided"
simps_2: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list" shows "I\<^sub>d\<^sub>l\<^sub>o (Less i j) xs = (xs ! i < xs ! j)"
moebius_circline_eq_I: fixes M :: "moebius"   and H1 :: "circline"   and H2 :: "circline" assumes "moebius_circline M H1 = moebius_circline M H2" shows "H1 = H2"
signed_cast_def: shows "signed_cast \<equiv> map_fun Word.rep Word.Word (signed_take_bit (LENGTH('k) - Suc 0))"
measure_loop_induct: fixes P :: "nat \<times> cell list \<times> cell list \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat \<times> cell list \<times> cell list" assumes "\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), f n) \<in> measure_loop" shows "\<exists>n. P (f n)"
lsc_hull_exists: fixes f :: "'b \<Rightarrow> ereal" shows "\<exists>g. Epigraph UNIV g = closure (Epigraph UNIV f)"
assoc_simps_5: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "cod \<a>[f, g, h] = cod f \<star> cod g \<star> cod h"
kim: shows "k ` sphere 0 1 \<subseteq> sphere b s"
convex_hull_mono: fixes Vs :: "'a vec set"   and Ws :: "'a vec set" assumes "Vs \<subseteq> Ws" shows "convex_hull Vs \<subseteq> convex_hull Ws"
expand: fixes tllist :: "('a, 'b) tllist"   and tllist' :: "('a, 'b) tllist" assumes "is_TNil tllist = is_TNil tllist'"   and "\<lbrakk>is_TNil tllist; is_TNil tllist'\<rbrakk> \<Longrightarrow> terminal tllist = terminal tllist'"   and "\<lbrakk>\<not> is_TNil tllist; \<not> is_TNil tllist'\<rbrakk> \<Longrightarrow> thd tllist = thd tllist' \<and> ttl tllist = ttl tllist'" shows "tllist = tllist'"
HCollectE: fixes a :: "hf"   and P :: "hf \<Rightarrow> bool"   and A :: "hf" assumes "a \<^bold>\<in> HCollect P A" obtains "a \<^bold>\<in> A" and "P a"
abs_eq: fixes x :: "'e \<Longrightarrow> 'f" shows "x = mk\<^sub>e get\<^bsub>x\<^esub>"
transUpdate_def: fixes a :: "'a"   and ec :: "'s odlist"   and ec' :: "'s odlist"   and at :: "('s, ('obs, 's odlist) mapping) trie" shows "SPRsingle.KBP.transUpdate a ec ec' at \<equiv> update trans_MapOps (ec, local.spr_simObs a ec') ec' at"
const_imply3: fixes P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'c"   and R :: "'a \<Rightarrow> 'd" assumes "\<And>\<tau> \<tau>'. \<lbrakk>P \<tau> = P \<tau>'; Q \<tau> = Q \<tau>'\<rbrakk> \<Longrightarrow> R \<tau> = R \<tau>'"   and "const P"   and "const Q" shows "const R"
chine_assoc_in_hom: shows "\<guillemotleft>chine_assoc : \<mu>\<nu>_\<pi>.chine \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.chine\<guillemotright>"
hyps: fixes A :: "(('b, 'c) vec, 'a) vec" assumes "n \<le> mod_type_class.to_nat i" shows "\<exists>P. invertible P \<and> Hermite_reduce_above A n i j res = P ** A"
iuf_2: shows "a2 \<in> fst ` set (routing_port_ranges tbl wordinterval_UNIV) \<and> b2 = wordinterval_Union (map snd (filter (\<lambda>x. a2 = fst x) (routing_port_ranges tbl wordinterval_UNIV)))"
list2policyR_End: fixes p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and list :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<notin> dom (Cp a)" shows "Cp (list2policyR (a # list)) p = (Cp a \<Oplus> list2policy (map Cp list)) p"
Norm: fixes G :: "prog"   and s :: "st" shows "G\<turnstile>Norm s \<midarrow>sxalloc\<rightarrow> Norm s"
less_annos_iff: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 < C2) = (C1 \<le> C2 \<and> (\<exists>i<length (annos C1). annos C1 ! i < annos C2 ! i))"
seq_char: fixes g :: "'a"   and f :: "'a" shows "R.seq g f = (R.arr f \<and> R.arr g \<and> seq g f)"
the_adj_chamber: fixes C :: "'a set"   and z :: "'a set" assumes "folding.chamber C"   and "z \<lhd> C" shows "folding.chamber (the_adj_chamber C z)"
separating_network_cut_of_sep: fixes S :: "('v \<times> 'v) set" assumes "separating \<Gamma> S"   and "source \<Delta> \<noteq> sink \<Delta>" shows "separating_network \<Delta> (fst ` \<E> S)"
effG_def: fixes eff' :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset option"   and r :: "'a"   and s :: "'b"   and sl :: "'b fset" shows "effG eff' r s sl \<equiv> RuleSystem_Code.eff eff' r s sl"
cinner_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c"   and r :: "complex"   and y :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (scaleC r y) = r * cinner x y"
Dirichlet_L_conv_eval_fds_weak: fixes s :: "complex" assumes "1 < Re s" shows "Dirichlet_L n \<chi>\<^sub>0 s = eval_fds (fds \<chi>\<^sub>0) s"
t1_space_nhds: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<forall>\<^sub>F x in topological_space.nhds open x. x \<noteq> y"
Diff_Un: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A - (B \<union> C) = (A - B) \<inter> (A - C)"
permute_Tree_tConj: fixes p :: "perm"   and tset :: "('b, 'c, 'd, 'e) Tree set['b]" shows "p \<bullet> tConj tset = tConj (p \<bullet> tset)"
map_tl: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f (tl xs) = tl (map f xs)"
atU_add_mset: fixes U :: "'a set"   and c :: "'a list"   and c2 :: "'a list multiset" shows "atU U (add_mset c c2) = (atU_s U c \<or> atU U c2)"
projecting_UNIV: fixes C :: "'d program \<Rightarrow> 'd set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program"   and X' :: "'d program set" shows "projecting C h F X' UNIV"
orderI: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "semilattice_order f less_eq less"   and "a = f a b" shows "less_eq a b"
lift_SD_efficient_iff: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set"   and p :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "alts \<noteq> {}"   and "agents \<noteq> {}"   and "finite alts'"   and "p \<in> lotteries_on alts" shows "SD_efficient (lift_pref_profile agents alts agents' alts' R) p = SD_efficient R p"
Re_Arcsin_bounds: fixes z :: "complex" shows "- pi < Re (Arcsin z) \<and> Re (Arcsin z) \<le> pi"
bit_of_bits_int_iff: fixes bs :: "bool list"   and n :: "nat" shows "bit (of_bits bs) n = nth_default (bs \<noteq> [] \<and> last bs) bs n"
simps_1: fixes \<theta> :: "trm"   and \<eta> :: "trm" shows "Pluso (Aterm \<theta>) (Aterm \<eta>) = Aterm (Plus \<theta> \<eta>)"
fresh_variable_type_scheme_lists: fixes A :: "type_scheme list" shows "\<exists>n. new_tv n A"
Some_Inf: fixes A :: "'b set" shows "Some (\<Sqinter> A) = \<Sqinter> (Some ` A)"
Imp_cut: fixes C :: "fm"   and H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "insert C H \<turnstile> A IMP B"   and "{A} \<turnstile> C" shows "H \<turnstile> A IMP B"
ctxt_2: shows "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"
Rep_matrix_mult: fixes a :: "'b matrix"   and b :: "'b matrix"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (a * b) j i = foldseq (+) (\<lambda>k. Rep_matrix a j k * Rep_matrix b k i) (max (ncols a) (nrows b))"
null_char: shows "B.VxVxV.null = (B.null, B.VxV.null)"
NSCauchyD: fixes X :: "nat \<Rightarrow> 'a"   and M :: "nat star"   and N :: "nat star" assumes "NSCauchy X"   and "M \<in> HNatInfinite"   and "N \<in> HNatInfinite" shows "(*f* X) M \<approx> (*f* X) N"
learn_total_closed_subseteq: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and V :: "(nat \<Rightarrow> nat option) set" assumes "learn_total \<psi> U s"   and "V \<subseteq> U" shows "learn_total \<psi> V s"
set_pmf_geometric: fixes p :: "real" assumes "0 < p"   and "p < 1" shows "set_pmf (geometric_pmf p) = UNIV"
quot_cases: fixes A :: "'d quot" obtains a :: "'d" where "A = \<lfloor>a\<rfloor>"
distinct_225: fixes x91 :: "Syntax.term"   and x92 :: "Syntax.term"   and x14 :: "Syntax.term" shows "App x91 x92 \<noteq> Unroll x14"
3: shows "A \<subseteq> (\<Union>y\<in>f ` A. A \<inter> f -` {y})"
continuous_map_imp_closed_graph: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "Hausdorff_space Y" shows "closedin (prod_topology X Y) ((\<lambda>x. (x, f x)) ` topspace X)"
usubstappf_geq_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" assumes "usubstappf \<sigma> U (Geq \<theta> \<eta>) \<noteq> undeff" shows "usubstappt \<sigma> U \<theta> \<noteq> undeft \<and> usubstappt \<sigma> U \<eta> \<noteq> undeft"
winning_region_extends_VVpstar: fixes v :: "'a"   and p :: "Player" assumes "v \<in> G'.VV p**"   and "\<And>w. v \<rightarrow>\<^bsub>G'\<^esub> w \<Longrightarrow> w \<in> G'.winning_region p" shows "v \<in> G'.winning_region p"
igSubstIGAbs_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIGAbs MOD \<equiv> \<forall>ys y Y xs x s X. isInBar (xs, s) \<and> igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
IH: shows "integral\<^sup>N (count_space UNIV) (h_plus i) \<noteq> \<top>"
advantage_elgamal: fixes \<eta> :: "nat"   and \<A> :: "('grp \<Rightarrow> (('grp \<times> 'grp) \<times> 'b) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "local.ind_cpa.advantage \<eta> \<A> = local.ddh.advantage \<eta> (elgamal_adversary \<eta> \<A>)"
sel: fixes r :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "the_gpv (bind_gpv r f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. bind_gpv r f)))) (bind_spmf (the_gpv r) (case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input))))))"
Q: shows "\<forall>u. u \<in> set Q \<longrightarrow> irreducible u \<and> monic u"
is_lasso_prpl_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and prpl :: "('Q \<times> nat) list \<times> ('Q \<times> nat) list" shows "gb_graph.is_lasso_prpl (b_graph.to_gbg_ext (degeneralize_ext ma) m) prpl \<equiv> graph.is_lasso_prpl_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) prpl \<and> (\<forall>A\<in>gbg_F (b_graph.to_gbg_ext (degeneralize_ext ma) m). set (snd prpl) \<inter> A \<noteq> {})"
pair_loopfree_digraph_axioms: shows "pair_loopfree_digraph (list_digraph G_list)"
tagged_division_Un: fixes p1 :: "('a \<times> 'a set) set"   and s1 :: "'a set"   and p2 :: "('a \<times> 'a set) set"   and s2 :: "'a set" assumes "p1 tagged_division_of s1"   and "p2 tagged_division_of s2"   and "interior s1 \<inter> interior s2 = {}" shows "p1 \<union> p2 tagged_division_of s1 \<union> s2"
a_g_x: shows "a \<succeq> x \<and> (x, a) \<notin> relation"
image_eqI: fixes b :: "'a::type"   and f :: "'b::type \<Rightarrow> 'a::type"   and x :: "'b::type"   and A :: "'b::type set" assumes "(b::'a::type) = (f::'b::type \<Rightarrow> 'a::type) (x::'b::type)"   and "(x::'b::type) \<in> (A::'b::type set)" shows "(b::'a::type) \<in> (f::'b::type \<Rightarrow> 'a::type) ` (A::'b::type set)"
Some_in_results'_gpv_catch_gpv: fixes x :: "'d"   and gpv :: "('d, 'e, 'f) gpv" shows "(Some x \<in> results'_gpv (catch_gpv gpv)) = (x \<in> results'_gpv gpv)"
I: shows "topspace X \<subseteq> \<Union> (T ` I)"
emeasure_PiM_emb: fixes J :: "'i set"   and X :: "'i \<Rightarrow> 'a set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M.events i" shows "emeasure (Pi\<^sub>M I M) (emb I J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (M i) (X i))"
E'_def: shows "E' \<equiv> \<^bold>E \<inter> X \<times> UNIV"
upd_rev_cs_Cons_Return: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" shows "CFG.upd_rev_cs kind cs (a # as) = a # CFG.upd_rev_cs kind cs as"
N_of_Liminf_state_inf: shows "N_of_state (Liminf_state Sts') \<subseteq> N_of_state (Liminf_state Sts)"
assms_2: fixes J :: "'a set" assumes "J \<in> P" shows "finite J"
abs_eq: fixes x :: "int" shows "Uint16 (integer_of_int x) = Abs_uint16 (word_of_int x)"
zero_vec_def: fixes n :: "nat" shows "0\<^sub>v n \<equiv> vec n (\<lambda>i. 0::'b)"
assms_3: fixes y :: "'a" assumes "y dvd a"   and "y dvd b" shows "y dvd x"
abs_sgn_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (sgn a) = (if a = zero then zero else one)"
isFDERIV_max_Var_congI: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list"   and ws :: "real list" assumes "isFDERIV n xs fas vs"   and "\<And>i. i < max_Var_floatariths fas \<Longrightarrow> vs ! i = ws ! i" shows "isFDERIV n xs fas ws"
normalize_simps_2: fixes f :: "'d \<Rightarrow> 'd" shows "Language.normalize (Basic f) = Basic f"
elims_1: fixes x :: "'c set"   and xa :: "('d, 'c) stateful_strand_step list"   and y :: "bool" assumes "wf'\<^sub>s\<^sub>s\<^sub>t x xa = y" obtains V :: "'c set" where "x = Vc__" and "xa = []" and "y"   | V :: "'c set" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vd__" and "xa = receive\<langle>ta__\<rangle> # Sb__" and "y = (fv ta__ \<subseteq> Vd__ \<and> wf'\<^sub>s\<^sub>s\<^sub>t Vd__ Sb__)"   | V :: "'c set" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Ve__" and "xa = send\<langle>tb__\<rangle> # Sc__" and "y = wf'\<^sub>s\<^sub>s\<^sub>t (Ve__ \<union> fv tb__) Sc__"   | V :: "'c set" and t :: "('d, 'c) Term.term" and t' :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vf__" and "xa = \<langle>tc__ := t'\<rangle> # Sd__" and "y = (fv t' \<subseteq> Vf__ \<and> wf'\<^sub>s\<^sub>s\<^sub>t (Vf__ \<union> fv tc__) Sd__)"   | V :: "'c set" and uu_ :: "('d, 'c) Term.term" and uv_ :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vg__" and "xa = \<langle>uu_ == uv_\<rangle> # Se__" and "y = wf'\<^sub>s\<^sub>s\<^sub>t Vg__ Se__"   | V :: "'c set" and t :: "('d, 'c) Term.term" and s :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vh__" and "xa = insert\<langle>td__,s__\<rangle> # Sf__" and "y = (fv td__ \<subseteq> Vh__ \<and> fv s__ \<subseteq> Vh__ \<and> wf'\<^sub>s\<^sub>s\<^sub>t Vh__ Sf__)"   | V :: "'c set" and uw_ :: "('d, 'c) Term.term" and ux_ :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vi__" and "xa = delete\<langle>uw_,ux_\<rangle> # Sg__" and "y = wf'\<^sub>s\<^sub>s\<^sub>t Vi__ Sg__"   | V :: "'c set" and t :: "('d, 'c) Term.term" and s :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vj__" and "xa = select\<langle>t,s\<rangle> # Sh__" and "y = wf'\<^sub>s\<^sub>s\<^sub>t (Vj__ \<union> fv t \<union> fv s) Sh__"   | V :: "'c set" and uy_ :: "('d, 'c) Term.term" and uz_ :: "('d, 'c) Term.term" and S :: "('d, 'c) stateful_strand_step list" where "x = Vk__" and "xa = \<langle>uy_ in uz_\<rangle> # Si__" and "y = wf'\<^sub>s\<^sub>s\<^sub>t Vk__ Si__"   | V :: "'c set" and va_ :: "'c list" and vb_ :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" and vc_ :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" and S :: "('d, 'c) stateful_strand_step list" where "x = V" and "xa = \<forall>va_\<langle>\<or>\<noteq>: vb_ \<or>\<notin>: vc_\<rangle> # S" and "y = wf'\<^sub>s\<^sub>s\<^sub>t V S"
w_simps_5: shows "local.dom \<omega>.chine = \<omega>.chine"
fset: fixes x :: "'b fset" shows "fset x \<in> {A. finite A}"
intT_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "intT \<sigma> a \<equiv> if unprot \<sigma> then eintT \<sigma> a else eintT \<sigma> a \<and> eintP (Guard \<sigma>) [a]"
set_entries_inject: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "(set (entries t1) = set (entries t2)) = (entries t1 = entries t2)"
SpanI_lincomb_arb_len_coeffs: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and rs :: "'a list"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "scalar_mult.lincomb smult rs ms \<in> R_scalar_mult.RSpan UNIV smult ms"
finite_set_pmf_of_list: fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs" shows "finite (set_pmf (pmf_of_list xs))"