hcju/isabelleps huggingface dataset:
Class_is_Block: fixes a :: "'a" assumes "a \<in> S" shows "Class a = Block a"
nprv_cut_set: fixes F :: "'fmla set"   and G :: "'fmla set"   and \<chi> :: "'fmla" assumes "finite F"   and "F \<subseteq> fmla"   and "finite G"   and "G \<subseteq> fmla"   and "\<chi> \<in> fmla"   and "\<And>\<psi>. \<psi> \<in> G \<Longrightarrow> nprv F \<psi>"   and "nprv (G \<union> F) \<chi>" shows "nprv F \<chi>"
face_im: fixes w :: "'a set"   and y :: "'c set"   and f :: "'a \<Rightarrow> 'c" assumes "w \<in> X"   and "order.greater_eq (f ` w) y" shows "y \<in> f \<turnstile> X"
infinite_remove: fixes S :: "'a set"   and a :: "'a" assumes "infinite S" shows "infinite (S - {a})"
complete2: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "A \<subseteq> B"   and "B \<in> null_sets (completion M)" shows "A \<in> null_sets (completion M)"
emeasure_eq_measure: fixes A :: "(int \<Rightarrow> 'a) set" shows "emeasure natural_extension_measure A = ennreal (I.prob A)"
abs_iso: fixes x :: "'a \<rightarrow> ('b \<times> 'a)\<cdot>'c" shows "stateT_rep\<cdot>(stateT_abs\<cdot>x) = x"
intros_3: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>"
abs_eq: fixes x :: "'b list" shows "pdevs_of_list x = Abs_pdev (\<lambda>i. if i < length x then x ! i else (0::'b))"
span_add_eq2: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "module scale"   and "y \<in> module.span scale S" shows "(x + y \<in> module.span scale S) = (x \<in> module.span scale S)"
connected_same: fixes u :: "'a"   and v :: "'a" shows "((u, v) \<in> connected {}) = (u = v \<and> v \<in> V)"
invar_tail_invar: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (t # bq)" shows "SkewBinomialHeapStruc.tail_invar bq"
interp_ops_unfold_last: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and x :: "'c \<times> ('c, 'd) list_op" shows "interp_ops (xs @ [x]) = interp_op (interp_ops xs) x"
reachable_snapshot_inv_def: fixes m :: "'d" shows "mut_m.reachable_snapshot_inv m = (\<lambda>s. \<forall>x. mut_m.reachable m x s \<longrightarrow> in_snapshot x s)"
assms: fixes A :: "'b trm"   and B :: "'b trm" shows "P \<noteq> SimplyTyped.Pair A B"
omax_in: fixes R :: "('c \<times> 'c) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Well_order r" shows "omax R \<in> R"
wf_syscls_class_Object: fixes P :: "'a prog" assumes "wf_syscls P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
limproc_is_thelub: fixes S :: "nat \<Rightarrow> 'c process" assumes "chain S" shows "Lub S = lim_proc (range S)"
is_const_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c"   and u :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_const unconst (app t u)"
prod_of_Zp_cauchy_is_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s"   and "is_Zp_cauchy t" shows "is_Zp_cauchy (s \<otimes>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> t)"
semilattice_axioms: shows "semilattice (\<sqinter>)"
in_set_zipE: fixes x :: "'c"   and y :: "'d"   and xs :: "'c list"   and ys :: "'d list" assumes "(x, y) \<in> set (zip xs ys)" obtains "x \<in> set xs" and "y \<in> set ys"
cproper_interval_char_def: shows "cproper_interval = proper_interval"
inj_on_add: fixes a :: "'a"   and A :: "'a set" shows "inj_on ((\<oplus>\<^sub>a) a) A"
regular_cover_def: fixes I :: "'c set"   and \<psi> :: "'c \<Rightarrow> ('a, 'b) chart" shows "regular_cover I \<psi> = (countable I \<and> carrier = (\<Union>i\<in>I. domain (\<psi> i)) \<and> locally_finite_on carrier I (domain \<circ> \<psi>) \<and> (\<forall>i\<in>I. codomain (\<psi> i) = ball (0::'b) 3) \<and> carrier = (\<Union>i\<in>I. inv_chart (\<psi> i) ` ball (0::'b) 1))"
extended_Gromov_product_e_x_x: fixes e :: "'a"   and x :: "'a Gromov_completion" shows "extended_Gromov_product_at e x x = extended_Gromov_distance (to_Gromov_completion e) x"
l1: fixes S :: "'node set"   and p :: "'node" assumes "\<And>p. p \<in> S \<Longrightarrow> \<exists>Q\<subseteq>S. quorum_of p Q"   and "p \<in> S" shows "quorum_of p S"
prems_1: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
equals_cemptyD: fixes A :: "'d cset"   and a :: "'d" assumes "A = cempty" shows "\<not> cin a A"
ptr_kinds_M_ptr_kinds: fixes h :: "(_) heap"   and xa :: "(_) object_ptr list" shows "h \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r xa = (xa = sorted_list_of_set (fset (object_ptr_kinds h)))"
affine_dependent_explicit: fixes p :: "'a set" shows "affine_dependent p = (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (0::'a))"
iprod_Nil: fixes ys :: "'a list" shows "\<langle>[],ys\<rangle> = (0::'a)"
cases: fixes x :: "nat \<times> nat" obtains x :: "nat" and y :: "nat" where "xa__ = (x, y)"
incrIndexList_help5: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "last (butlast ls) < nmax - Suc 0"
berlekamp_resulting_mat_closed_3: fixes u :: "'c mod_ring poly" shows "dim_col (berlekamp_resulting_mat u) = degree u"
similar_mat_wit_hom: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" assumes "similar_mat_wit A B C D" shows "similar_mat_wit (mat\<^sub>h A) (mat\<^sub>h B) (mat\<^sub>h C) (mat\<^sub>h D)"
add_eq_conv_diff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and N :: "'b multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (remove1_mset a N) \<and> N = add_mset a (remove1_mset b M))"
simps_1: fixes x1 :: "('c, 'd) insert_message"   and y1 :: "('c, 'd) insert_message" shows "(Insert x1 = Insert y1) = (x1 = y1)"
intro: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
key_gen_def: shows "key_gen = G \<bind> (\<lambda>(x, w). return_spmf (x, x, w))"
linear_surjective_imp_injective: fixes f :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "surj f"   and "dim UNIV = dim UNIV" shows "inj f"
s'A_m: fixes n :: "nat" shows "swaps (paid_A ! n) (s_A n) = s'_A n (length (paid_A ! n))"
set_coeffs_subset_singleton_0_iff: fixes p :: "'a poly" shows "(set (coeffs p) \<subseteq> {0::'a}) = (p = 0)"
double_csin_square: fixes a :: "real" shows "2 * complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos (2 * a))"
eval_elim_cases_7: fixes G :: "prog"   and s :: "st"   and w :: "val"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Lit w)\<succ>\<rightarrow> (v, s')" obtains "v = In1 w" and "s' = Norm s"
ms_of_greek_inv_greek: fixes M :: "(accent \<times> 'a) list" shows "ms_of_greek (inv_greek M) = inv_msog (ms_of_greek M)"
counit_is_iso: shows "local.iso \<epsilon>\<^sub>0"
stable_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list"   and p :: "nat" shows "stable r step \<tau>s p = (\<forall>(q, \<tau>)\<in>set (step p (\<tau>s ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q)"
strict_monoD: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "strict_mono f"   and "x < y" shows "f x < f y"
borel_measurable_implies_sequence_metric: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and z :: "'d" assumes "f \<in> borel_measurable M" shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) \<longlonglongrightarrow> f x) \<and> (\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
select: fixes \<C> :: "t_context"   and t :: "t" shows "\<C> \<turnstile> [Select] : [t, t, T_i32] _> [t]"
while_proof: fixes Q :: "'S \<Rightarrow> bool" assumes "\<And>s. \<lbrakk>s \<in> dwa_invar WA; \<not> dwa_cond WA s\<rbrakk> \<Longrightarrow> Q s" shows "Q loop"
simps_1: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (floatarith.Add a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (add_aform' p a1 a2)))"
inj_split_Cons: fixes X :: "('a list \<times> 'a) set" shows "inj_on (\<lambda>(xs, n). n # xs) X"
simps_5: fixes r :: "'b mrexp3"   and s :: "'b mrexp3"   and f1 :: "'b set"   and n :: "bool" shows "ok3 (Times3 r s f1 n) = (ok3 r \<and> ok3 s \<and> (let rs = Times (strip3 r) (strip3 s) in f1 = Collect (final1 rs) \<and> n = nullable rs))"
word_eqI: fixes u :: "'a word"   and v :: "'a word" assumes "\<And>n. n < size u \<longrightarrow> u !! n = v !! n" shows "u = v"
PiE_cong: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
fold_set_fold: fixes y :: "'a pred"   and xs :: "'a pred list" shows "Finite_Set.fold (\<squnion>) y (set xs) = fold (\<squnion>) xs y"
that: fixes y :: "'a" assumes "periodic_orbit y"   and "range (flow0 y) = \<omega>_limit_set x" shows "thesis"
SUP_upper: fixes i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd \<times> 'e" assumes "i \<in> A" shows "less_eq_prod (f i) (Sup_prod (f ` A))"
simps_7: fixes x3 :: "error"   and x1 :: "xcpt" shows "Error x3 \<noteq> Xcpt x1"
llength_inf_llist: fixes f :: "nat \<Rightarrow> 'a" shows "llength (inf_llist f) = \<infinity>"
Pi_def: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
sup_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (sup x y) z = sup (times x z) (times y z)"
poly_root_unity: fixes n :: "nat"   and x :: "'a" shows "(poly (root_unity n) x = (0::'a)) = (x ^ n = (1::'a))"
fmrestrict_fset_dom: fixes m :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmrestrict_fset (fmdom m) m = m"
sums_unique: fixes f :: "nat \<Rightarrow> 'd"   and s :: "'d" assumes "f sums s" shows "s = suminf f"
ccTTree_mono1: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" assumes "S \<subseteq> S'" shows "ccTTree S G \<sqsubseteq> ccTTree S' G"
max_rank_of_def: fixes \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<Sigma> :: "'b set set"   and \<psi> :: "'b ltl" shows "ltl_to_rabin_base_def.max_rank_of \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> \<psi> \<equiv> semi_mojmir_def.max_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<psi>))"
1: assumes "degree q \<noteq> 0" shows "sturm_aux (real_of_rat_poly q) (real_of_rat_poly (- (p mod q))) = map real_of_rat_poly (sturm_aux_rat q (- (p mod q)))"
post_mult_beta: shows "post *\<^sub>v \<beta> = \<beta>"
D_compE\<^sub>1: fixes e :: "char list exp"   and Vs :: "char list list" assumes "\<D> e \<lfloor>set Vs\<rfloor>"   and "fv e \<subseteq> set Vs"   and "distinct Vs" shows "\<D> (compE\<^sub>1 Vs e) \<lfloor>{..<length Vs}\<rfloor>"
simps: fixes out :: "'b list"   and nexts :: "(nat \<times> 'b) list"   and prevs :: "(nat \<times> 'b) list"   and out' :: "'b list"   and head' :: "'c"   and tail' :: "'d"   and nexts' :: "(nat \<times> 'b) list" shows "next_subseqs1_spec out nexts prevs (out', head', tail', nexts') = (set nexts' = set nexts \<union> next_subseqs1_set prevs \<and> set out' = set out \<union> out_subseqs1_set prevs)"
rPar2: fixes \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>C. Prop C \<Psi> (!P) \<alpha> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop C \<Psi> (P \<parallel> !P) \<alpha> (P \<parallel> P')"
fresh_2: fixes a :: "name"   and x1 :: "name" shows "a \<sharp> Var x1 = a \<sharp> x1"
prod_uminus: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Prod>x\<in>A. - f x) = (- (1::'a)) ^ card A * prod f A"
mset_subset_eqI: fixes A :: "'k multiset"   and B :: "'k multiset" assumes "\<And>a. count A a \<le> count B a" shows "A \<subseteq># B"
removeNones_split: shows "removeNones vol = hd (removeNones vol) # tl (removeNones vol)"
Lim_component_le_cart: fixes f :: "'a \<Rightarrow> (real, 'b) vec"   and l :: "(real, 'b) vec"   and net :: "'a filter"   and i :: "'b"   and b :: "real" assumes "(f \<longlongrightarrow> l) net"   and "net \<noteq> bot"   and "\<forall>\<^sub>F x in net. f x $ i \<le> b" shows "l $ i \<le> b"
funct_14: fixes m00 :: "'a"   and m01 :: "'a"   and m10 :: "'a"   and m11 :: "'a"   and c0 :: "bool"   and c1 :: "bool" shows "funct_OT_14 (m00, m01, m10, m11) (c0, c1) = return_spmf ((), if c0 then if c1 then m11 else m10 else if c1 then m01 else m00)"
enabled_eff': fixes eff' :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset option"   and r :: "'a"   and s :: "'b" shows "i.enabled eff' r s = (eff' r s \<noteq> None)"
in_diff_countE: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<in># M - N" obtains n :: "nat" where "count M x = Suc n + count N x"
vimage_right_total_transfer: fixes B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique B"   and "right_total A" shows "((A ===> B) ===> rel_set B ===> rel_set A) (\<lambda>f X. f -` X \<inter> Collect (Domainp A)) (-`)"
sfirst_finite: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(sfirst P \<omega> < \<infinity>) = ev P \<omega>"
right_mult_zero_mat: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n" shows "A * 0\<^sub>m n nc = 0\<^sub>m nr nc"
fold_middle_sum_chain: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and as :: "'a permutation list"   and s :: "'a permutation"   and bs :: "'a permutation list"   and t :: "'a permutation"   and cs :: "'a permutation list" assumes "(f, g) \<in> foldpairs"   and "as \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "bs \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "t \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "cs \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "sum_list as `\<rightarrow> C0 \<in> f \<turnstile> \<C>"   and "sum_list (as @ [s]) `\<rightarrow> C0 \<in> g \<turnstile> \<C>"   and "sum_list (as @ [s] @ bs) `\<rightarrow> C0 \<in> g \<turnstile> \<C>"   and "sum_list (as @ [s] @ bs @ [t]) `\<rightarrow> C0 \<in> f \<turnstile> \<C>" shows "sum_list (as @ [s] @ bs @ [t] @ cs) `\<rightarrow> C0 = sum_list (as @ bs @ cs) `\<rightarrow> C0"
set_intros_2: fixes x1 :: "'a" shows "x1 \<in> vars_hd (Var x1)"
transp_onI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" shows "transp_on P A"
a1: shows "X \<or>\<^sub>A C \<Longrightarrow>\<^sub>t D"
Chi_def: fixes A :: "('a, 'b, 'c) hierauto" shows "Chi A \<equiv> \<lambda>S\<in>HAStates A. {S'. \<exists>SA\<in>SAs A. SA \<in> the (CompFun A S) \<and> S' \<in> States SA}"
m_comm: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and f :: "'a" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "ProblemIkTpart.TE_resOf resOf (Oldf f) = resOf f"
guha_matched: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and fe :: "('a, 'c) flow_entry_match"   and p :: "'b"   and ft1 :: "('a, 'c) flow_entry_match list"   and ft2 :: "('a, 'c) flow_entry_match list" assumes "\<gamma> (ofe_fields fe) p = True"   and "\<forall>fe'\<in>set (ft1 @ ft2). ofe_prio fe < ofe_prio fe' \<longrightarrow> \<gamma> (ofe_fields fe') p = False" shows "guha_table_semantics \<gamma> (ft1 @ fe # ft2) p (Some (ofe_action fe))"
eq': shows "t \<^sup>@ i \<cdot> r \<cdot> u \<cdot> q = t \<^sup>@ k"
closed_map_paired_continuous_map_left: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "Hausdorff_space Y" shows "closed_map X (prod_topology Y X) (\<lambda>x. (f x, x))"
finite_fset: fixes S :: "'b fset" shows "finite (fset S)"
enabled_eff': fixes r :: "'rule"   and s :: "'state" shows "enabled r s = (eff' r s \<noteq> None)"
assms_4: shows "val_Zp a \<le> val_Zp b"
has_finite_products': fixes I :: "'a set" assumes "I \<noteq> UNIV"   and "finite I" shows "has_products I"
remove_key_sum: fixes k :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" shows "remove_key k f + Poly_Mapping.single k (lookup f k) = f"
pos_literal_in_imp_true_cls: fixes A :: "'b"   and C :: "'b literal multiset"   and I :: "'b set" assumes "Pos A \<in># C"   and "A \<in> I" shows "I \<Turnstile> C"
mor_image1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g" shows "f ` B1 \<subseteq> B1'"
decode_plan_step_element_then_i: fixes k :: "nat"   and t :: "nat"   and \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "k < t" shows "set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k) = {\<Pi>\<^sub>\<O> ! index (\<Pi>\<^sub>\<O>) op |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op))}"
ffb_prop: fixes f :: "'c \<Rightarrow> 'b set" shows "fb\<^sub>\<F> f = \<partial> \<circ> bd\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
weaken: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<hookrightarrow> \<Psi> \<otimes> \<Psi>'"
lens_put_of_quotient: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e"   and s :: "'e"   and v\<^sub>2 :: "'d"   and v\<^sub>1 :: "'f" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "put\<^bsub>Y\<^esub> s (put\<^bsub>X /\<^sub>L Y\<^esub> v\<^sub>2 v\<^sub>1) = put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> s v\<^sub>2) v\<^sub>1"
cont_While_True: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'c"   and s :: "'b"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('c, 'd, 'a) cmd"   and i :: "nat" assumes "PL cval"   and "tval tst s = True" shows "PL.cont aval tval cval (While tst c) s i = c ;; While tst c"
less_imp_le: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "a \<preceq>\<^bsub>D\<^esub> b"
simps_6: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and A :: "'a tree"   and B :: "'a tree"   and CD :: "'a tree" assumes "x < a"   and "x < b"   and "A \<noteq> \<langle>\<rangle>" shows "splay x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle> = (case splay x A of \<langle>A1, a', A2\<rangle> \<Rightarrow> \<langle>A1, a', \<langle>A2, a, \<langle>B, b, CD\<rangle>\<rangle>\<rangle>)"
sup_eq_cases: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<squnion> - y = - x \<squnion> - z"   and "- - x \<squnion> - y = - - x \<squnion> - z" shows "- y = - z"
hurwitz_zeta_formula: assumes "1 \<in> {0<..1}"   and "s \<noteq> 0"   and "1 \<noteq> 1 \<or> s \<noteq> 1" shows "rGamma s * hurwitz_zeta 1 (1 - s) = complex_of_real (2 * pi) powr - s * (\<i> powr - s * perzeta 1 s + \<i> powr s * perzeta (- 1) s)"
ded_right: fixes a :: "'b G"   and b :: "'b G"   and c :: "'b G" shows "a * b r\<rightarrow> c = b r\<rightarrow> a r\<rightarrow> c"
cases: fixes x :: "complex \<times> complex \<times> complex" obtains k :: "complex" and xa :: "complex" and y :: "complex" where "x = (k, xa, y)"
P2: shows "P2 = P1(i := mktop (P1 i) b)"
sum_mult_Tr1: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R" shows "(\<forall>j\<le>n. f j \<in> set_mult R A B) \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> carrier R"
varsC_substC: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and c :: "('c, 'd) lit list" shows "varsC (substC \<pi> c) = (\<Union>x\<in>varsC c. vars (\<pi> x))"
assms: shows "\<forall>j\<le>2 * d. str (t + j + Suc (i + k)) = x"
preserves_def: fixes v :: "'d \<Rightarrow> 'e" shows "preserves v \<equiv> \<Inter>z. stable {s. v s = z}"
zero_change_imp_all_preconds_submap: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap"   and b :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s' :: "('a, 'b) fmap" assumes "vars_change as vs s = []"   and "sat_precond_as s as"   and "ListMem b as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "fmrestrict_set vs (fst b) \<subseteq>\<^sub>f fmrestrict_set vs s'"
causal_program_history: fixes sb\<^sub>1 :: "'a memref list"   and sb\<^sub>2 :: "'a memref list" assumes "sb = sb\<^sub>1 @ sb\<^sub>2" shows "\<exists>is. instrs sb\<^sub>2 @ Ghost A L R W # is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2"
simps_15: fixes va :: "nat" shows "\<infinity>\<^sub>1 +\<^sub>1 [Suc va]\<^sub>1 = \<top>\<^sub>1"
card_insert_le: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "card A \<le> card (insert x A)"
countableI': fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "inj_on f S" shows "countable S"
subset_def: fixes A :: "ZF"   and B :: "ZF" shows "HOLZF.subset A B \<equiv> \<forall>x. Elem x A \<longrightarrow> Elem x B"