hcju/isabelleps huggingface dataset:
aHom_inv_inv: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F" shows "f (-\<^sub>a\<^bsub>F\<^esub> a) = -\<^sub>a\<^bsub>G\<^esub> f a"
wt_env_mono: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and T :: "ty"   and E' :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> e :: T"   and "E \<subseteq>\<^sub>m E'" shows "P,E' \<turnstile> e :: T"
closure_of_topspace: fixes X :: "'a topology" shows "X closure_of topspace X = topspace X"
\<delta>_statesI_2: fixes q :: "'c"   and l :: "'d"   and qs :: "'c list"   and \<delta> :: "('c, 'd) ta_rule set" assumes "q \<rightarrow> l qs \<in> \<delta>" shows "set qs \<subseteq> \<delta>_states \<delta>"
Rep_uint_inverse: fixes x :: "uint" shows "Abs_uint (Rep_uint x) = x"
leg_simps_1: shows "C.arr \<mu>.dom.leg0"
assms_2: shows "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"
eq_numeral_iff_iszero_7: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus x) = one) = ring_1.iszero zero (numeral.numeral one plus (x + num.One))"
a_consistent_let: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, Terms.Let \<Delta> e, S)"   and "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S"   and "edom ae \<inter> domA \<Delta> = {}" shows "a_consistent (Aheap \<Delta> e\<cdot>a \<squnion> ae, a, as) (\<Delta> @ \<Gamma>, e, S)"
vset_proj_imp_vset: fixes a :: "('d, bool) fmap \<times> ('d, bool) fmap"   and vs' :: "'d set"   and vs :: "'d set" assumes "varset_action a vs'"   and "varset_action (action_proj a vs') vs" shows "varset_action a vs"
enum_strict_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(enum i < enum j) = (i < j)"
simps: fixes R :: "('b \<times> 'a) set"   and a :: "'b" shows "R ,, a = the_elem (R `` {a})"
bounded_iff: fixes b :: "'c \<times> 'd"   and c :: "'c \<times> 'd"   and a :: "'c \<times> 'd" shows "less_eq_prod (sup_prod b c) a = (less_eq_prod b a \<and> less_eq_prod c a)"
monoid_axioms: shows "Group_Theory.monoid R (\<cdot>) \<one>"
nrows_mult: fixes A :: "'b matrix"   and B :: "'b matrix" shows "nrows (A * B) \<le> nrows A"
REFPHI: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma>' :: "'\<sigma>a"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R" shows "\<Phi> it \<sigma>"
wt_intF: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "list_all2 intT (arOf f) al" shows "intT (resOf f) (intF f al)"
inject_2: fixes x21 :: "'a"   and x22 :: "'a form list"   and y21 :: "'a"   and y22 :: "'a form list" shows "(Compound x21 x22 = Compound y21 y22) = (x21 = y21 \<and> x22 = y22)"
Node: fixes q :: "('a, 'b) pq" shows "set (pq.alist_of q) \<subseteq> set (pq.alist_of (bts_augment rs q))"
path_connected_punctured_universe: fixes a :: "'a" assumes "2 \<le> DIM('a)" shows "path_connected (- {a})"
simps_3: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 \<Colon> x22 \<noteq> []"
valid_typeII_div: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "valid_typeII_division s two_chain"
satur: shows "FL.empty_ord.saturated (Liminf_llist (lmap lclss_of_state Sts))"
ttl_tmap: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and xs :: "('i, 'j) tllist" shows "ttl (tmap f g xs) = tmap f g (ttl xs)"
sset_sconst: fixes x :: "'a" shows "sset (sconst x) = {x}"
blinding_of_on_def: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "blinding_of_on A h bo \<equiv> blinding_respects_hashes h bo \<and> blinding_of_on_axioms A bo"
weakTransitionI: fixes P :: "('a, 'b, 'c) psi"   and P'' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''"   and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P'') \<Psi>"   and "\<Psi> \<rhd> P'' \<longmapsto> \<alpha> \<prec> P'" shows "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
min_int_poly_def: fixes x :: "'a" shows "min_int_poly x = (if algebraic x then THE p. p represents x \<and> irreducible p \<and> 0 < Polynomial.lead_coeff p else [:0, 1:])"
Hurwitz_no_zeros: fixes S :: "complex set"   and \<F> :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex"   and z0 :: "complex" assumes "open S"   and "connected S"   and "\<And>n. \<F> n holomorphic_on S"   and "g holomorphic_on S"   and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"   and "\<not> g constant_on S"   and "\<And>n z. z \<in> S \<Longrightarrow> \<F> n z \<noteq> 0"   and "z0 \<in> S" shows "g z0 \<noteq> 0"
dvd_add_triv_right_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b a) = dvd.dvd times a b"
atLeastLessThan_Suc: fixes a :: "nat"   and b :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<le> b" shows "prod g {a..<Suc b} = prod g {a..<b} * g b"
delete_edges_wf: fixes G :: "'c list_graph"   and E :: "('c \<times> 'c) list" assumes "wf_list_graph G" shows "wf_list_graph (FiniteListGraph.delete_edges G E)"
beukers_denom_neq: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x \<in> {0<..<1}"   and "y \<in> {0<..<1}"   and "z \<in> {0..1}" shows "(1 - x * y) * z \<noteq> 1"
weak_transition_tau_iff: fixes P :: "'state"   and P' :: "'state" shows "P \<Rightarrow>\<langle>\<tau>\<rangle> P' = P \<Rightarrow> P'"
gs_6: shows "compl_fam (\<lambda>g. generate G {g}) (set gs)"
cosh_square_eq: fixes x :: "'a" shows "(cosh x)\<^sup>2 = (sinh x)\<^sup>2 + (1::'a)"
member_empty: shows "DList_Set.member DList_Set.empty = (\<lambda>_. False)"
weaken_left_Un2: fixes G :: "'b pl set"   and p :: "'b pl"   and A :: "'b pl set" assumes "G |- p" shows "A \<union> G |- p"
monoid_axioms: shows "monoid (\<cdot>) (1::'a)"
map_dual_dual: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(\<partial>\<^sub>F f = g) = (\<partial>\<^sub>F g = f)"
field_has_derivative_at: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c" shows "(f has_derivative (*) D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'c\<rightarrow> D"
diag_mat_def: fixes f :: "'d \<Rightarrow> 'c" shows "diag_mat f = (\<chi>i j. if i = j then f i else (0::'c))"
atLeastAtMost_iff: fixes i :: "'a \<times> ('a, 'b) operation"   and l :: "'a \<times> ('a, 'b) operation"   and u :: "'a \<times> ('a, 'b) operation" shows "(i \<in> hb.atLeastAtMost l u) = (weak_hb l i \<and> weak_hb i u)"
at_least_at_most_closed: fixes L :: "('a, 'b) gorder_scheme"   and a :: "'a"   and b :: "'a" assumes "weak_partial_order L" shows "\<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> \<subseteq> carrier L"
max_step: fixes j :: "nat" assumes "t j \<le> t1" shows "max_stepsize j \<le> \<bar>r\<bar> * L / B / (exp (L * (t1 - t 0) + 1) - 1)"
cas_subset: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and H :: "('c, 'd) pre_digraph" assumes "pre_digraph.cas G u p v"   and "subgraph G H" shows "pre_digraph.cas H u p v"
less_top_Top_is_Value: fixes x :: "'b top" assumes "x < top.Top" shows "\<exists>z. x = top.Value z"
risk_neutral_param: fixes q :: "real" assumes "q = (1 + r - d) / (u - d)"   and "viable_market Mkt" shows "u * q + d * (1 - q) = 1 + r"
0: shows "Fvars \<chi> = Fvars \<phi> - {x} \<union> (if x \<in> Fvars \<phi> then FvarsT t else {})"
intro: fixes CC :: "('c, 'b, 'd) category_scheme" assumes "\<And>f. f \<in> Ar\<^bsub>CC\<^esub> \<Longrightarrow> Dom\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"   and "\<And>f. f \<in> Ar\<^bsub>CC\<^esub> \<Longrightarrow> Cod\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"   and "\<And>f. f \<in> Ar\<^bsub>CC\<^esub> \<Longrightarrow> Id\<^bsub>CC\<^esub> (Cod\<^bsub>CC\<^esub> f) \<bullet>\<^bsub>CC\<^esub> f = f"   and "\<And>f. f \<in> Ar\<^bsub>CC\<^esub> \<Longrightarrow> f \<bullet>\<^bsub>CC\<^esub> Id\<^bsub>CC\<^esub> (Dom\<^bsub>CC\<^esub> f) = f"   and "\<And>A. A \<in> Ob\<^bsub>CC\<^esub> \<Longrightarrow> Id\<^bsub>CC\<^esub> A \<in> Hom\<^bsub>CC\<^esub> A A"   and "\<And>A B C. (\<bullet>\<^bsub>CC\<^esub>) \<in> Hom\<^bsub>CC\<^esub> B C \<rightarrow> Hom\<^bsub>CC\<^esub> A B \<rightarrow> Hom\<^bsub>CC\<^esub> A C"   and "\<And>f g h. \<lbrakk>f \<in> Ar\<^bsub>CC\<^esub>; g \<in> Ar\<^bsub>CC\<^esub>; h \<in> Ar\<^bsub>CC\<^esub>; Cod\<^bsub>CC\<^esub> h = Dom\<^bsub>CC\<^esub> g; Cod\<^bsub>CC\<^esub> g = Dom\<^bsub>CC\<^esub> f\<rbrakk> \<Longrightarrow> f \<bullet>\<^bsub>CC\<^esub> (g \<bullet>\<^bsub>CC\<^esub> h) = f \<bullet>\<^bsub>CC\<^esub> g \<bullet>\<^bsub>CC\<^esub> h" shows "category CC"
card_greater_0_if_post_after_pre: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list" assumes "i < j"   and "postrecording_event t i"   and "prerecording_event t j" shows "0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}"
either_sym: fixes t :: "'b ttree"   and t2 :: "'b ttree" shows "t \<oplus>\<oplus> t2 = t2 \<oplus>\<oplus> t"
Cons_1: fixes accu :: "('a \<times> 'a) list" assumes "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu (list_graph_to_graph G) (get_spec M) accu Es"
true_cls_def: fixes I :: "'b set"   and C :: "'b literal multiset" shows "(I \<Turnstile> C) = (\<exists>L\<in>#C. I \<Turnstile>l L)"
b_assn_subtyping_match_eqA_tR: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" shows "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt A x y"
mono_SUP: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'a \<Rightarrow> 'b"   and A :: "'c \<Rightarrow> 'a"   and I :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "(SUP x\<in>I. f (A x)) \<le> f (Sup (A ` I))"
PolysD_2: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "p \<in> P[X]" shows "indets p \<subseteq> X"
SeqAbstFormP_Disj: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, x, y, x', y', v, i)"   and "atom k \<sharp> (s1, s2, k1, k2, x, y, x', y', v, i)" shows "{SeqAbstFormP v i x x' s1 k1, SeqAbstFormP v i y y' s2 k2, OrdP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i (Q_Disj x y) (Q_Disj x' y') (Var s) (Var k)))"
d_OUT_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "incseq (\<lambda>n y. f n (x, y))" shows "d_OUT (\<lambda>e. \<Squnion>n. f n e) x = (\<Squnion>n. d_OUT (f n) x)"
set_disconnected_nodes_get_parent_is_l_set_disconnected_nodes_get_parent: shows "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"
simps_1: fixes y :: "'a arctic_delta" shows "MinInfty_delta + y = y"
merge_ask: fixes A :: "'s"   and f :: "'a \<Rightarrow> 'r \<Rightarrow> ('a, 'm) nondetT" shows "merge A (\<lambda>x. ask (\<lambda>r. run_nondet (f x r))) = ask (\<lambda>r. merge A (\<lambda>x. run_nondet (f x r)))"
fv_body: shows "fv body \<subseteq> set (this # pns)"
diff_ge_0_iff_ge: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
card_partition_on_size1: fixes A :: "'b set" assumes "finite A" shows "card {P. partition_on A P \<and> (\<forall>X\<in>P. card X = 1)} = 1"
base_in_hom_2: shows "\<guillemotleft>r \<star> s : r \<star> s \<Rightarrow> r \<star> s\<guillemotright>"
sifum_security_init_axioms: shows "sifum_security_init dma \<C>_vars eval\<^sub>w"
dvdnumcoeff_trans: fixes g :: "int"   and g' :: "int"   and t :: "Ferrack.num" assumes "g dvd g'"   and "dvdnumcoeff t g'" shows "dvdnumcoeff t g"
append_eqvt: fixes p :: "perm"   and xs :: "'a list"   and ys :: "'a list" shows "p \<bullet> (xs @ ys) = p \<bullet> xs @ p \<bullet> ys"
BAD_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "BAD P = {f \<in> SEQ A. bad P f}"
en_eq_1: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<diamond>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>x,F\<rbrace> in v]"
synth_trans: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> synth G"   and "G \<subseteq> synth H" shows "X \<in> synth H"
lincomb_pad_1: fixes U :: "'a vec set"   and n :: "nat"   and a :: "'a vec \<Rightarrow> 'a"   and m :: "nat" assumes "U \<subseteq> carrier_vec n"   and "finite U" shows "lincomb n a U @\<^sub>v 0\<^sub>v m = lincomb (n + m) (a \<circ> unpadr m) ((\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` U)"
Bfun_const: fixes c :: "'c"   and F :: "'b filter" shows "Bfun (\<lambda>_. c) F"
mult2_alt_nsI: fixes X :: "'b multiset"   and X1 :: "'b multiset"   and X2 :: "'b multiset"   and Y :: "'b multiset"   and Y1 :: "'b multiset"   and Y2 :: "'b multiset"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "X = X1 + X2"   and "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)" shows "(X, Y) \<in> mult2_alt_ns ns s"
zero_root_unit: fixes n :: "nat" shows "(root_unity n = 0) = (n = 0)"
iterate_to_list_map_rev_linord_correct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "(('a \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_rev_linord less_eq it (map_to_set m)" shows "map_of (iterate_to_list it) = m \<and> distinct (map fst (iterate_to_list it)) \<and> linorder.sorted less_eq (map fst (iterate_to_list it))"
onl_oinvariant_sterms: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and A :: "('e \<times> ('b, 'c, 'a, 'd) seqp, 'f) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<times> 'd \<Rightarrow> bool"   and \<sigma> :: "'e"   and p :: "('b, 'c, 'a, 'd) seqp"   and p' :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d" assumes "wellformed \<Gamma>"   and "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<sigma>, l)"
pderiv_gen_shleg_poly: fixes n :: "nat"   and k :: "nat" shows "pderiv (gen_shleg_poly n k) = gen_shleg_poly n (Suc k)"
Gromov_product_nonneg: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "0 \<le> Gromov_product_at e x y"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'c"   and u :: "'c"   and y :: "'c"   and v :: "'c" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
is_final: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n1 :: "nat"   and n2 :: "nat" assumes "is_final (steps c p n1)"   and "n1 \<le> n2" shows "is_final (steps c p n2)"
VDM_Sound_emptyCtxt: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "{} \<rhd> c : A" shows " \<Turnstile> c : A "
arity_def: fixes s :: "('s, 'v) tm" shows "arity s = arity_hd (head s) - enat (num_args s)"
cut_cut': fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a antiparallel_edges.vertex set" assumes "antiparallel_edges \<Delta>"   and "MFMC_Network.cut (antiparallel_edges.\<Delta>'' \<Delta>) S" shows "MFMC_Network.cut \<Delta> (antiparallel_edges.Vertex -` S)"
f_one_to_one_on_index_subset_2: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and Y :: "'a set"   and y :: "'a"   and k :: "nat" assumes "[f[f 0 ..]X]"   and "0 \<le> i"   and "i < j"   and "Y = f ` {i..j}"   and "y \<in> Y" shows "f k = y \<longrightarrow> k \<in> {i..j}"
caseWeakSimPres: fixes CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and Eq :: "'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool"   and \<Psi> :: "'b"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> Eq \<Psi> P Q"   and "\<And>\<Psi>' P Q. (\<Psi>', P, Q) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> P \<leadsto><Rel> Q"   and "\<And>\<Psi>' P Q. Eq \<Psi>' P Q \<Longrightarrow> (\<Psi>', P, Q) \<in> Rel"   and "\<And>\<Psi>' P Q. Eq \<Psi>' P Q \<Longrightarrow> \<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q" shows "\<Psi> \<rhd> Cases CsP \<leadsto><Rel> Cases CsQ"
r_dec_def: shows "r_dec \<equiv> Cn 1 (Pr 1 Z (recf.Id 3 0)) [recf.Id 1 0, recf.Id 1 0]"
p_not_sys: fixes p :: "'a process_name" shows "(p \<noteq> sys) = (p = gc \<or> (\<exists>m. p = mutator m))"
Compl_set_less_eq_aux_Compl: fixes A :: "'a set"   and B :: "'a set" assumes "finite UNIV" shows "(- A \<sqsubseteq>' - B) = (B \<sqsubseteq>' A)"
cart2_append1_def: fixes p :: "proj2" shows "cart2_append1 p \<equiv> (1 / proj2_rep p $ 3) *\<^sub>R proj2_rep p"
c_offending: shows "c_offending_flows m = (\<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP)"
mprefix_singl: fixes a :: "'a"   and P :: "'a \<Rightarrow> 'a process" shows "Mprefix {a} P = (a \<rightarrow> P a)"
coef_apps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm"   and ss :: "('b, 'a) tm list"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.coef coef_sym (apps s ss) i = tkbo_coefs.coef coef_sym s (i + length ss)"
reindex_cong: fixes l :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
dist_triangle2: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist x z + dist y z"
r_combine_non_Leaves: fixes t :: "'b tree"   and u :: "'b tree" assumes "t \<noteq> Leaf"   and "u \<noteq> Leaf" shows "r (combine t u) = r u"
distrib_lattice_def: fixes 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" shows "class.distrib_lattice inf less_eq less sup \<equiv> class.lattice inf less_eq less sup \<and> class.distrib_lattice_axioms inf sup"
Fun_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)" shows "Fun (mkIde A) = (\<lambda>x\<in>A. x)"
phl_skip: fixes x :: "'a" shows "x \<cdot> (1::'a) \<le> (1::'a) \<cdot> x"
Inf_insert: fixes S :: "real set"   and x :: "real" assumes "bounded S" shows "Inf (insert x S) = (if S = {} then x else min x (Inf S))"
wellformed_policy3_charn: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "singleCombinators p"   and "distinct p"   and "allNetsDistinct p"   and "wellformed_policy1 p"   and "wellformed_policy2 p" shows "wellformed_policy3 p"
wf_interp_for_formula_FAnd: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi>1 :: "'b formula"   and \<phi>2 :: "'b formula" assumes "formula \<Sigma>" shows "formula.wf_interp_for_formula \<Sigma> (w, I) (FAnd \<phi>1 \<phi>2) = (formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>1 \<and> formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>2)"
thetaSAtm_Sretr: fixes atm :: "'atom" assumes "compatAtm atm" shows "thetaSAtm atm \<subseteq> Sretr (thetaSAtm atm)"
Basic: fixes a :: "'a set"   and A :: "'a set set"   and sp :: "'a set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
not_strong_postdominate_predecessor_successor: fixes n :: "'node"   and a :: "'edge" assumes "\<not> n strongly-postdominates sourcenode a"   and "valid_node n"   and "valid_edge a" obtains a' :: "'edge" where "valid_edge a'" and "sourcenode a' = sourcenode a" and "\<not> n strongly-postdominates targetnode a'"
is_functor: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map"