hcju/isabelleps huggingface dataset:
elims_1: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and y :: "bool" assumes "pred x xa = y" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>f__" and "xa = cf__ # cfs__" and "y"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Qa__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "y = Qa__ (fst cfa__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qb__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "y = Qb__ (fst cfb__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "y = (Q cf \<and> cfs \<noteq> [])"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" where "x = et" and "xa = []" and "\<not> y"
project_set_extend_set_Un: fixes A :: "'a set"   and B :: "'c set" shows "project_set h (extend_set h A \<union> B) = A \<union> project_set h B"
open_path_connected_component_set: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "path_component_set S x = connected_component_set S x"
antisym: fixes a :: "'b"   and b :: "'b" assumes "codomain.greater_eq b a"   and "codomain.greater_eq a b" shows "a = b"
abs_convergent_prod_conv_summable: fixes f :: "nat \<Rightarrow> 'e" shows "abs_convergent_prod f = summable (\<lambda>i. norm (f i - (1::'e)))"
hd_filter_prop: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" assumes "filter P xs \<noteq> []" shows "P (hd (filter P xs))"
Aenc_Enc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Aenc X Y \<noteq> Enc X' Y'"
build_rel_def: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
set_compre_diff: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "{x \<in> A - B. P x} = {x \<in> A. P x} - {x \<in> B. P x}"
lea_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "A B C LeA F E D"
globally_consistent_adapt_bisim: fixes c\<^sub>1 :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "globally_consistent A mds mem\<^sub>1 mem\<^sub>2" shows "\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>"
not_Cons_self: fixes xs :: "'a list"   and x :: "'a" shows "xs \<noteq> x # xs"
fun_upd_upd: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'g"   and z :: "'g" shows "f(x := y, x := z) = f(x := z)"
valid_W_invD0_4: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name"   and q :: "'g process_name" assumes "r \<in> ghost_honorary_grey (s p)"   and "valid_W_inv s"   and "p \<noteq> q" shows "r \<notin> WL q s"
not_iszero_neg_1: shows "\<not> iszero (- (1::'a))"
ex_psi_i_f: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> U" shows "\<exists>i. \<psi> i = f"
AE_all_accD: fixes P :: "'c cfg \<Rightarrow> bool"   and s :: "'c cfg"   and s' :: "'c cfg" assumes "almost_everywhere (MC.T s) (pred_stream P)"   and "(s, s') \<in> MC.acc" shows "almost_everywhere (MC.T s') (pred_stream P)"
l5_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "Bet A B C"   and "Bet A B D" shows "Bet A C D \<or> Bet A D C"
and_funct_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "and_funct A B = (let (a1, a2) = A; (b1, b2) = B in coin_spmf \<bind> (\<lambda>\<sigma>. return_spmf (\<sigma>, \<sigma> \<oplus> (a1 \<oplus> b1 \<and> a2 \<oplus> b2))))"
sgn1_is_pr: shows "sgn1 \<in> PrimRec1"
has_derivative_add: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') F"   and "(g has_derivative g') F" shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
ord_eq_le_trans: fixes a :: "'b::type"   and b :: "'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and c :: "'b::type" assumes "(a::'b::type) = (b::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (b::'b::type) (c::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (c::'b::type)"
PowI: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "A \<in> Pow B"
growth_rate_euclidean_below: fixes R :: "real"   and D :: "real" assumes "0 \<le> R" shows "\<exists>F\<subseteq>cball (0::'b) R. (\<forall>x\<in>F. \<forall>y\<in>F. x = y \<or> D \<le> dist x y) \<and> finite F \<and> (1 / (max D 1 * real DIM('b))) ^ DIM('b) * R ^ DIM('b) \<le> real (card F)"
comm_monoid_mult_hom_1_axioms: shows "comm_monoid_mult_hom_1 of_real"
hfref_to_ASSERT_conv: fixes P :: "'f \<Rightarrow> bool"   and a :: "'g \<Rightarrow> 'h Heap"   and b :: "'f \<Rightarrow> 'i nres"   and R :: "('f \<Rightarrow> 'g \<Rightarrow> assn) \<times> ('f \<Rightarrow> 'g \<Rightarrow> assn)"   and S :: "'i \<Rightarrow> 'h \<Rightarrow> assn" assumes "NO_MATCH (\<lambda>_. True) P" shows "((a, b) \<in> [P]\<^sub>a R \<rightarrow> S) = ((a, \<lambda>x. ASSERT (P x) \<bind> (\<lambda>_. b x)) \<in> R \<rightarrow>\<^sub>a S)"
cases: fixes x :: "bool \<times> nat \<times> int vec list" obtains upw :: "bool" and i :: "nat" and fs :: "int vec list" where "x = (upw, i, fs)"
prv_imp_com: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<phi> (imp \<chi> \<psi>))" shows "bprv (imp \<chi> (imp \<phi> \<psi>))"
VerduennungListeS: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<Theta>" shows "\<Gamma> \<Rightarrow> \<Theta> @ \<Theta>"
domain_permutation_def: fixes A :: "'b set"   and B :: "'c set" shows "domain_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p. p permutes A \<and> (\<forall>x\<in>A. f x = f' (p x)))}"
refine2spec_aux: fixes a :: "'a nres"   and R :: "('a \<times> 'b) set"   and b :: "'b nres" shows "(a \<le> \<Down> R b) = (nofail b \<longrightarrow> a \<le> SPEC (\<lambda>r. \<exists>x. inres b x \<and> (r, x) \<in> R))"
trivial_ker_imp_inj: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h"   and "kernel G H h = {\<one>\<^bsub>G\<^esub>}" shows "inj_on h (carrier G)"
append1_eq_conv: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and y :: "'b" shows "(xs \<cdot> [x] = ys \<cdot> [y]) = (xs = ys \<and> x = y)"
a1: shows "\<nabla>x = (zero_class.zero::'a)"
assms_1: shows "F \<simeq>\<^sub>F G"
list_disj_def: fixes fs :: "'c fm list" shows "list_disj fs = foldr or fs FalseF"
mult_oner: fixes x :: "'a ppath set" shows "pp_prod x pp_one = x"
length_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less" shows "length (linorder.sorted_list_of_set less_eq A) = card A"
subint: fixes I :: "qtname"   and J :: "qtname"   and G :: "prog" assumes "G\<turnstile>I\<preceq>I J" shows "G\<turnstile>Iface I\<preceq>Iface J"
dom_in_domains: fixes f :: "'s" assumes "domains f \<noteq> {}" shows "local.dom f \<in> domains f"
simps_8: fixes v :: "Formula.formula"   and va :: "Formula.formula" shows "remove_neg (formula.And v va) = formula.And v va"
Theap_substitute: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "ttree_restr (domA \<Delta>) (substitute (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (thunks \<Delta>) (Texp e\<cdot>a)) \<sqsubseteq> Theap \<Delta> e\<cdot>a"
allocationsUniverseOutside: fixes a :: "('c \<times> 'd set) set"   and X :: "'c set" assumes "a \<in> allocationsUniverse" shows "Outside' X a \<in> allocationsUniverse"
cn0: shows "lookup p (t \<oplus> lt f) \<noteq> (0::'b)"
assms_1: shows "ordermap A r y \<in> elts (ordermap A r x)"
S1_def: shows "S1 = {y. x - y \<in> supp f \<and> y \<in> supp g}"
mult_col_div_row_carrier_3: fixes A :: "'b mat"   and n :: "nat"   and a :: "'b"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "mult_col_div_row a k A \<in> carrier_mat n n"
ka_k_or_Suc: shows "ka = k \<or> ka = Suc k"
starfun_exp_HInfinite: fixes x :: "real star" assumes "x \<in> HInfinite"   and "0 \<le> x" shows "(*f* exp) x \<in> HInfinite"
AbstractTransform_axioms: shows "AbstractTransform (Rep_cfun inc) (Rep_cfun pred) (\<lambda>\<Delta> e a. (a, Aheap \<Delta> e\<cdot>a)) fst snd (\<lambda>_. 0) (\<lambda>a. Var) (\<lambda>a. App) (\<lambda>a. Lam) (\<lambda>b \<Gamma>. Terms.Let (map_transform Aeta_expand (snd b) \<Gamma>))"
Bot_delR: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta> - (\<bottom>, {#})"
simpleDefs_phiDefs_disjoint: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> step.phiDefs g n = {}"
r_def: shows "R = P1 - {Pos A} \<union> (P2 - {Neg A})"
not_is_PureE: fixes generat :: "('g, 'h, 'i) generat" assumes "\<not> is_Pure generat" obtains out :: "'h" and c :: "'i" where "generat = IO out c"
test_compl_1: fixes x :: "'a" assumes "is_test x" shows "x + tc x = 1'"
sing_pow_palindrom: fixes w :: "'b list"   and a :: "'b"   and k :: "nat" assumes "w = [a] \<^sup>@ k" shows "rev w = w"
routing_action_oiface_update_def: fixes h :: "char list"   and pk :: "'b routing_rule" shows "routing_action_oiface_update h pk = routing_action_update (output_iface_update (\<lambda>_. h)) pk"
MX: shows "map_minsky (\<lambda>s. if s = 0 then 3 else 2 * s) id M \<subseteq> X"
char_one: shows "\<chi> \<one>\<^bsub>G\<^esub> = 1"
\<eta>\<epsilon>_props: shows "adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"
simps_1: fixes v :: "nat"   and s :: "nat"   and s' :: "nat" shows "encode_Op (Dec v s s') = c_pair 0 (c_pair v (c_pair s s'))"
elims: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and xa :: "'a option"   and xb :: "'b option"   and y :: "assn" assumes "option_assn x xa xb = y" obtains P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" where "x = P__" and "xa = None" and "xb = None" and "y = emp"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and a :: "'a" and c :: "'b" where "x = P" and "xa = Some a" and "xb = Some c" and "y = P a c"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'a" where "x = uu___" and "xa = Some v__" and "xb = None" and "y = false"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'b" where "x = uu_" and "xa = None" and "xb = Some v" and "y = false"
is_OK_returns_heap_E: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog" assumes "h \<turnstile> ok f" obtains h' :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>h h'"
\<phi>_def: shows "\<phi> = (\<lambda>c d. c = h ::= Ct 1 ;; l ::= Ct 2 \<and> d = l ::= Ct 2 \<or> d = h ::= Ct 1 ;; l ::= Ct 2 \<and> c = l ::= Ct 2)"
finsum_all0: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a vec" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = 0\<^sub>v n" shows "finsum V f A = 0\<^sub>v n"
lm011: fixes N :: "'a set"   and G :: "'b set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "N \<noteq> {}"   and "finite N"   and "G \<noteq> {}"   and "finite G" shows "winningAllocationRel N G (\<noteq>) bids {} \<and> winningAllocationRel N G finite bids"
atms_empty_iff_empty: fixes C :: "'b literal multiset" shows "(atms_of C = {}) = (C = {#})"
correct_fpc: fixes xs :: "pratt list"   and p :: "nat"   and qs :: "nat list"   and r :: "nat"   and a :: "nat" assumes "valid_cert xs"   and "1 < p"   and "prod_list qs = r"   and "r \<noteq> 0"   and "\<forall>q\<in>set qs. Prime q \<in> set xs"   and "\<forall>q\<in>set qs. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)" shows "valid_cert (build_fpc p a r qs @ xs)"
good_imp_qGood_pick: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X" shows "qGood (pick X)"
prems: shows "all_roots_real ([:- x, 1:] * p)"
freeIn_altNegInf: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (alternateNegInfinity p var)"
vector_export_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> top \<sqinter> y \<sqinter> z = x \<sqinter> top \<sqinter> (y \<sqinter> z)"
iTILL_iIN_subset: fixes n' :: "nat"   and d :: "nat" assumes "n' \<le> d" shows "[\<dots>n'] \<subseteq> [0\<dots>,d]"
Limit: fixes S :: "((('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real) \<times> (('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real)) set" assumes "\<forall>x\<in>S. P (fst x) (snd x)"   and "\<forall>x\<in>S. feasible (fst x)"   and "\<forall>x\<in>S. \<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)" shows "P (Sup_trans (fst ` S)) (Inf_utrans (snd ` S))"
less_return_time_imp_exivl: fixes h :: "real"   and P :: "'a set"   and x' :: "'a" assumes "h \<le> Y.return_time P x'"   and "Y.returns_to P x'"   and "closed P"   and "0 \<le> h" shows "h \<in> Y.existence_ivl0 x'"
Cons_1: fixes v1 :: "'a \<times> nat" assumes "SubRel.subpath g v1 es v2 subs"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
countable_SIGMA: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (Sigma I A)"
gamma3_y_const: shows "\<forall>x. gamma3 x \<bullet> j = b"
disjointD: fixes A :: "'a set set"   and a :: "'a set"   and b :: "'a set" assumes "disjoint A"   and "a \<in> A"   and "b \<in> A"   and "a \<noteq> b" shows "a \<inter> b = {}"
lemma3_5_d_v: fixes ars :: "('c \<times> 'd \<times> 'c) set"   and \<tau> :: "'c \<times> ('d \<times> 'c) list"   and \<sigma> :: "'c \<times> ('d \<times> 'c) list"   and \<sigma>' :: "'c \<times> ('d \<times> 'c) list"   and \<tau>' :: "'c \<times> ('d \<times> 'c) list"   and \<upsilon> :: "'c \<times> ('d \<times> 'c) list"   and \<upsilon>' :: "'c \<times> ('d \<times> 'c) list"   and \<tau>'' :: "'c \<times> ('d \<times> 'c) list" assumes "diagram ars (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "diagram ars (\<tau>', \<upsilon>, \<upsilon>', \<tau>'')" shows "diagram ars (\<tau>, (fst \<sigma>, snd \<sigma> @ snd \<upsilon>), (fst \<sigma>', snd \<sigma>' @ snd \<upsilon>'), \<tau>'')"
fin_make_untimed_inf_truncate_Nonempty: fixes z :: "nat \<Rightarrow> 'd list"   and k :: "nat"   and i :: "nat" assumes "z k \<noteq> []"   and "k \<le> i" shows "fin_make_untimed (inf_truncate z i) \<noteq> []"
matrix_scalar_mult_def: fixes k :: "'b"   and A :: "(('b, 'c) vec, 'd) vec" shows "k *k A \<equiv> \<chi>i j. k * A $ i $ j"
psitau_in_P2: shows "\<psi>\<^sup>\<tau> \<in> \<P>\<^sup>2"
HDomain_Incl_SeqAppendP: fixes f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and k :: "tm" shows "{SeqAppendP f1 k1 f2 k2 g, HDomain_Incl f1 k1 AND HDomain_Incl f2 k2, HaddP k1 k2 k, OrdP k1} \<turnstile> HDomain_Incl g k"
create_gba_from_nodes_precond_axioms: shows "create_gba_from_nodes_precond \<phi> qs"
drop_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "drop n (drop m xs) = drop (n + m) xs"
changes_append: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs = hd ys \<and> last xs \<noteq> (0::'a)" shows "changes (xs @ ys) = changes xs + changes ys"
prime_elem_power_dvd_prod: fixes p :: "'a"   and c :: "nat"   and m :: "'a"   and n :: "'a" assumes "p ^ c dvd m * n"   and "prime_elem p" shows "\<exists>a b. a + b = c \<and> p ^ a dvd m \<and> p ^ b dvd n"
a_get_ancestors_locs_def: fixes get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs = get_parent_locs"
hn: assumes "w1 \<noteq> POS_INF \<and> 0 <s w1 \<and> w2 = NEG_INF" shows "P w1 w2"
ext_inject: fixes view :: "'d \<Longrightarrow> 'f"   and coview :: "'e \<Longrightarrow> 'f"   and more :: "'g"   and view' :: "'d \<Longrightarrow> 'f"   and coview' :: "'e \<Longrightarrow> 'f"   and more' :: "'g" shows "(\<lparr>view = view, coview = coview, \<dots> = more\<rparr> = \<lparr>view = view', coview = coview', \<dots> = more'\<rparr>) = (view = view' \<and> coview = coview' \<and> more = more')"
surjective: fixes r :: "('a, 'b, 'c) rec_scheme" shows "r = \<lparr>rec1 = rec1 r, rec2 = rec2 r, \<dots> = more r\<rparr>"
valid_frac_def: fixes X :: "('a \<times> 'a) set" shows "qr.valid_frac X \<equiv> \<exists>r\<in>R. \<exists>s\<in>R\<setminus>\<pp>. qr.frac r s = X"
ro: shows "a \<in> read_only \<S> \<longrightarrow> a \<in> read_only \<S>'"
star_invol: fixes x :: "'b \<times> 'c" shows "prod.qstar (prod.qstar x) = prod.qstar x"
assms_2: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
assms: fixes x :: "'a" shows "summable (\<lambda>n. c n * x ^ n)"
\<R>\<^sub>3_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'__ = c\<^sub>1''__ ;; c__" and "\<langle>c\<^sub>1''__, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>''__,\<S>''__,Pb__\<^esub> \<langle>c\<^sub>2''__, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>''__,\<S>''__,Pb__ {c__} \<Gamma>'__,\<S>'__,Pa__" and "c\<^sub>2'__ = c\<^sub>2''__ ;; c__" and "mds\<^sub>2 = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'__ = c\<^sub>1 ;; c" and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>'__,\<S>'__,Pa__" and "c\<^sub>2'__ = c\<^sub>2 ;; c" and "mds\<^sub>2 = mds"
new_def: fixes p :: "('c, 'd) fm"   and a :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list" shows "new p a branch \<equiv> \<not> p at a in branch"
C_eq_until_separated: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l))) = C p"
one_over_poly_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "one_over_poly (monom P a n) = monom P a 0"
sup_ge1: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
free_vars_cexpr_subst_val_aux: fixes x :: "nat"   and e :: "cexpr"   and v :: "val" shows "free_vars (cexpr_subst_val_aux x e v) = (\<lambda>y. if x \<le> y then Suc y else y) -` free_vars e"
HComp_int_def: shows "(\<otimes>) \<equiv> map_fun rep_interface (map_fun rep_interface abs_interface) (\<otimes>\<^sub>c)"
assms_2: fixes m :: "'a" assumes "R \<sigma> m" shows "R \<sigma>' m"
evret_flow: shows "\<forall>\<^sub>F x in at x0 within S. returns_to p2 (flow0 x (return_time p1 x))"
find_closest_pair_set: fixes C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" assumes "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps" shows "C\<^sub>0 \<in> set ps \<and> C\<^sub>1 \<in> set ps \<or> C\<^sub>0 = c\<^sub>0 \<and> C\<^sub>1 = c\<^sub>1"
vector_smult_assoc: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "a *s (b *s x) = a * b *s x"
trace_def: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" shows "trace n f g xs x \<equiv> map (\<lambda>y. the (eval (Pr n f g) (y # xs))) (rev [0..<Suc x])"
check_no_overlap_def: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list" shows "check_no_overlap \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. \<forall>p\<in>UNIV. ofe_prio a = ofe_prio b \<and> \<gamma> (ofe_fields a) p \<and> a \<noteq> b \<longrightarrow> \<not> \<gamma> (ofe_fields b) p)"
max_d'_lt_a: assumes "D' \<noteq> {#}" shows "Max (atms_of D') < A"
times_rel_fun_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" shows "f \<star> g = \<otimes> \<rho> f g"
merge_index: fixes I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'b option list"   and tJ :: "'b option list"   and t :: "'b option list"   and i :: "nat" assumes "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ"   and "t = New_max.merge tI tJ"   and "i < n" shows "i \<in> I \<and> t ! i = tI ! i \<or> i \<in> J \<and> t ! i = tJ ! i \<or> i \<notin> I \<and> i \<notin> J \<and> t ! i = None"
Pow_def: fixes A :: "'b set" shows "Pow A = {B. order.greater_eq A B}"
assms: shows "x - y = 2 * real_of_int k * pi \<or> x + y = 2 * real_of_int k * pi"
simps_3: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d' = \<lbrakk>insert t M; S\<rbrakk>\<^sub>d'"
getIJProperties_1: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" assumes "2 \<le> card V"   and "(I, J) = getIJ Q_pos Q_neg V" shows "1 \<le> card I"
maxU34: shows "wmax (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2) \<equiv>\<^sub>U max (rl1 * ru2) (ru1 * ru2)"
simps_8: fixes \<mu> :: "'a ltln"   and \<mu>' :: "'a ltln" shows "pure_eventual (\<mu> M\<^sub>n \<mu>') = (pure_eventual \<mu> \<and> pure_eventual \<mu>' \<or> \<mu>' = true\<^sub>n)"