hcju/isabelleps huggingface dataset:
cong_exp_iff_simps_10: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = False"
not_in_segment1: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "x \<notin> segment f x y"
mult: fixes u :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "simple_function M u"   and "P u" shows "P (\<lambda>x. c * u x)"
ln_minus_ln_floor_bound: fixes x :: "real" assumes "2 \<le> x" shows "ln x - ln (real_of_int \<lfloor>x\<rfloor>) \<in> {0..<1 / (x - 1)}"
insert_Diff_zmset: fixes x :: "'b"   and M :: "'b zmultiset" shows "add_zmset x (M - {#x#}\<^sub>z) = M"
weaken_asm_Un: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "bool"   and B :: "'b set" assumes "(\<forall>x\<in>A. P x) \<longrightarrow> Q" shows "(\<forall>x\<in>A \<union> B. P x) \<longrightarrow> Q"
cong_exp_iff_simps_4: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus m) (numeral.numeral one plus num.One) = modulo (numeral.numeral one plus n) (numeral.numeral one plus num.One)) = True"
Sup_non_neg: shows "0 \<le> Sup {\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> = 1}"
image_diff_atMost: fixes n :: "nat" shows "(-) n ` {..n} = {..n}"
permutep_single: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'c" assumes "bij f" shows "permutep f (Poly_Mapping.single a b) = Poly_Mapping.single (inv f a) b"
size_mirror: fixes t :: "'c tree" shows "size (mirror t) = size t"
open_ball: fixes x :: "'c"   and d :: "real" shows "open {y. dist x y < d}"
NEGc_simps_2: fixes C :: "ty" shows "\<parallel><NOT C>\<parallel> = AXIOMSc (NOT C) \<union> BINDINGc (NOT C) (\<parallel>(NOT C)\<parallel>) \<union> NOTRIGHT (NOT C) (\<parallel>(C)\<parallel>)"
sum_of_two_squares': fixes n :: "nat" shows "is_sum2sq_nat n = (\<forall>p. prime p \<and> [p = 3] (mod 4) \<longrightarrow> even (multiplicity p n))"
subcringE_6: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h1 :: "'c"   and h2 :: "'c" assumes "subcring H R"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"
fpxs_shift_add: fixes r :: "rat"   and f :: "'b fpxs"   and g :: "'b fpxs" shows "fpxs_shift r (f + g) = fpxs_shift r f + fpxs_shift r g"
three_neq_1: shows "\<^bold>1 \<noteq> \<^bold>2"
schroeder_6_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (id z \<sqinter> x \<le> - id y)"
fh: fixes x :: "'a" assumes "x \<in> T" shows "f x = exp (h x)"
length_dims0_minus_vars1'_is_vars2': shows "{0..<length dims0} - vars1' = {0..<length dims0} \<inter> vars2'"
hcomplex_hRe_one: shows "hRe 1 = 1"
prod_prime_factors: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "comm_monoid_mult.prod times one (\<lambda>p. power.power one times p (factorial_semiring.multiplicity times one p x)) (set_mset (factorial_semiring.prime_factorization zero times one normalize x)) = normalize x"
continuous_open_imp_quotient_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "open_map X X' f"   and "f ` topspace X = topspace X'" shows "quotient_map X X' f"
cos_eq_minus1: fixes x :: "real" shows "(cos x = - 1) = (\<exists>xa. x = (2 * real_of_int xa + 1) * pi)"
alw_shift: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xl :: "'d list"   and xs :: "'d stream" assumes "alw \<phi> (xl @- xs)" shows "alw \<phi> xs"
in_cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> O[F](h)) = (g \<in> O[F](h))"
simps: fixes a1 :: "(bool list option \<times> unit \<times> bool list cstate) spmf"   and a2 :: "(bool list option \<times> bool list cstate) spmf" shows "local.S a1 a2 = (a1 = return_spmf (None, (), Void) \<and> a2 = return_spmf (None, Void) \<or> (\<exists>plain. a1 = return_spmf (None, (), Store plain) \<and> a2 = map_spmf (\<lambda>key. (Some key, Store (key [\<oplus>] plain))) (spmf_of_set (nlists UNIV \<eta>)) \<and> length plain = id' \<eta>) \<or> (\<exists>plain. a1 = return_spmf (None, (), cstate.Collect plain) \<and> a2 = map_spmf (\<lambda>key. (Some key, cstate.Collect (key [\<oplus>] plain))) (spmf_of_set (nlists UNIV \<eta>)) \<and> length plain = id' \<eta>) \<or> (\<exists>key plain. a1 = return_spmf (Some (key [\<oplus>] plain), (), Store plain) \<and> a2 = return_spmf (Some key, Store (key [\<oplus>] plain)) \<and> length plain = id' \<eta> \<and> length key = id' \<eta>) \<or> (\<exists>key plain. a1 = return_spmf (Some (key [\<oplus>] plain), (), cstate.Collect plain) \<and> a2 = return_spmf (Some key, cstate.Collect (key [\<oplus>] plain)) \<and> length plain = id' \<eta> \<and> length key = id' \<eta>) \<or> a1 = return_spmf (None, (), cstate.Fail) \<and> a2 = map_spmf (\<lambda>x. (Some x, cstate.Fail)) (spmf_of_set (nlists UNIV \<eta>)) \<or> (\<exists>key plain. a1 = return_spmf (Some (key [\<oplus>] plain), (), cstate.Fail) \<and> a2 = return_spmf (Some key, cstate.Fail) \<and> length plain = id' \<eta> \<and> length key = id' \<eta>))"
dim_eq_hyperplane: fixes S :: "'a set" shows "(dim S = DIM('a) - 1) = (\<exists>a. a \<noteq> (0::'a) \<and> span S = {x. a \<bullet> x = 0})"
wlt_leadsTo: fixes F :: "'c program"   and B :: "'c set" shows "F \<in> wlt F B \<longmapsto> B"
remove_unis_sentence: fixes p :: "('a, 'b) form"   and m :: "nat" assumes "infinite (- params p)"   and "closed 0 (put_unis m p)"   and "[] \<turnstile> put_unis m p" shows "[] \<turnstile> p"
simps_2: fixes uu :: "'b" shows "bn (Eff uu) = {}"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_in_subst: fixes p :: "'a strand_label \<times> ('b, 'c) Term.term \<times> ('b, 'c) Term.term"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and \<delta> :: "'c \<Rightarrow> ('b, 'c) Term.term" assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>q\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. fst p = fst q \<and> (\<exists>X\<subseteq>bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. snd p = snd q \<cdot>\<^sub>p rm_vars X \<delta>)"
list_all2': shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ (A_x \<union> A_y) # A_xs @ A_ys) @ A_neg) ((zs @ join x True y # xs @ ys) @ L_neg)"
dis: shows "disjnt C (\<Union> (\<G> \<inter> Pow V))"
fresh_pj: fixes a :: "atom"   and j :: "name" assumes "a \<sharp> p"   and "j \<in> Vs" shows "a \<sharp> p \<bullet> j"
nonz: shows "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0"
igWlsAbsDisj_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gWlsAbsDisj MOD" shows "igWlsAbsDisj (fromMOD MOD)"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
reciprocal_of_complex: fixes z :: "complex" assumes "z \<noteq> 0" shows "reciprocal (of_complex z) = of_complex (1 / z)"
intro_elim_4_d: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and \<Theta> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]"   and "[\<phi> \<^bold>\<rightarrow> \<chi> in v]"   and "[\<psi> \<^bold>\<rightarrow> \<Theta> in v]" shows "[\<chi> \<^bold>\<or> \<Theta> in v]"
set_bit_def: fixes n :: "nat"   and a :: "'a" shows "set_bit n a = a OR push_bit n (1::'a)"
3: shows "cod\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = cod\<^bsub>C\<^esub> g"
list_rel_simp_2: fixes l :: "'e list"   and R :: "('e \<times> 'f) set" shows "((l, []) \<in> \<langle>R\<rangle>list_rel) = (l = [])"
distinct_3: fixes x1 :: "nat" shows "(_ x1 _) \<noteq> (_Exit_)"
nop_vdomain: shows "\<D>\<^sub>\<circ> vint_plus = \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
that: fixes u :: "'a list"   and a :: "'a"   and v :: "'a list"   and w' :: "'a stream" assumes "w = u @- a ## v @- a ## w'" shows "thesis"
HMSet_eq_0_iff: fixes m :: "hmultiset multiset" shows "(HMSet m = 0) = (m = {#})"
invar1: shows "prim_invar1 (A Q \<pi>)"
g'_def: shows "local.dis_log_alt.g' = \<^bold>g [^] x"
assms: shows "A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = B1 @ B2"
bst_baliL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "Tree2.bst l"   and "Tree2.bst r"   and "\<forall>x\<in>Tree2.set_tree l. x < a"   and "\<forall>x\<in>Tree2.set_tree r. a < x" shows "Tree2.bst (baliL l a r)"
simps_1: fixes pu :: "'a \<Rightarrow> bool"   and ar :: "'a \<Rightarrow> nat"   and M :: "('a, 'b) Term.term set"   and x :: "'b" shows "intruder_synth' pu ar M (Var x) = (Var x \<in> M)"
lookup_a1_map_keys': shows "\<forall>xs\<in>Mapping.keys a1_map'. case Mapping.lookup a1_map' xs of Some tp' \<Rightarrow> tp' < tp + 1"
union_grounding_of_cls_ground: fixes N :: "'a literal multiset set" shows "is_ground_clss (\<Union> (\<G>_F ` N))"
right_v1: shows "right (len v1 ts c) = left (ext v2)"
cauchy_filter_metricI: fixes F :: "'c filter" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>P. eventually P F \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist x y < e)" shows "cauchy_filter F"
sorted_wrt_append: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "List.sorted_wrt P (xs @ ys) = (List.sorted_wrt P xs \<and> List.sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
Nset_inc_0: fixes n :: "nat" shows "0 \<in> {i. i \<le> n}"
a: shows "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"
new_document_no_child_nodes: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr) \<rightarrow>\<^sub>r []"
distinct_5: fixes x1 :: "'a"   and x41 :: "bexp"   and x42 :: "'a acom"   and x43 :: "'a acom"   and x44 :: "'a" shows "SKIP {x1} \<noteq> IF x41 THEN x42 ELSE x43 {x44}"
zero: fixes f :: "'d \<Rightarrow> 'e" assumes "Modules.additive f" shows "f (0::'d) = (0::'e)"
h1: shows "\<forall>t. length (a t) \<le> Suc 0"
1: fixes x :: "('a, bool) fmap list" assumes "x \<in> state_set (state_list s as)" shows "last x \<in> reachable_s PROB s"
simps_2: fixes d :: "nat" shows "intv_const (Intv d) = d"
simps_4: fixes K :: "nat"   and X :: "freemsg" shows "freeleft (DECRYPT K X) = freeleft X"
measure_spmf_zero_iff: fixes p :: "'a spmf"   and A :: "'a set" shows "(Sigma_Algebra.measure (measure_spmf p) A = 0) = (set_spmf p \<inter> A = {})"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1"
snd_add: fixes x :: "'d \<times> 'c"   and y :: "'d \<times> 'c" shows "snd (x + y) = snd x + snd y"
constant_function_closed: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "\<cc>\<^bsub>Zp\<^esub> a \<in> carrier (function_ring (carrier Zp) Zp)"
integrable_on_scaleR_left: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and y :: "'d" assumes "f integrable_on A" shows "(\<lambda>x. f x *\<^sub>R y) integrable_on A"
s1_s2_P1: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and xc :: "nat"   and y :: "nat" assumes "x < q"   and "xa < q"   and "xb < q"   and "xc < q"   and "y < q" shows "((x, xa, xb, (y + q - xc) mod q), (x * ((y + q - xc) mod q) + q - xb) mod q, ((x + xa) mod q * xc + q - (xa * xc + q - xb) mod q) mod q) = ((x, xa, xb, (y + q - xc) mod q), (x * ((y + q - xc) mod q) + q - xb) mod q, (x * y + q - (x * ((y + q - xc) mod q) + q - xb) mod q) mod q)"
set_map_idx: fixes f :: "'d \<Rightarrow> nat \<Rightarrow> 'c"   and xs :: "'d list"   and n :: "nat" shows "set (map_idx f xs n) = (\<lambda>i. f (xs ! i) (i + n)) ` {0..<length xs}"
map_of_None2: fixes ps :: "('a \<times> 'b) list"   and x :: "'a" assumes "Sorted_Less.sorted (map fst ps @ [x])" shows "AList_Upd_Del.map_of ps x = None"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_inverse" shows "a\<inverse>\<^sub>\<real> = b"
FunctorComp'_def: fixes F :: "('a, 'g, 'c, 'h, 'e, 'i) Functor"   and G :: "('g, 'b, 'h, 'd, 'j, 'f) Functor" shows "F ;;: G \<equiv> \<lparr>CatDom = CatDom F, CatCod = CatCod G, MapM = \<lambda>f. G ## F ## f\<rparr>"
L0: shows "Infer (zip (prems_of \<iota>) Ll) (concl_of \<iota>, L0) \<in> Inf_FL"
Le_le_LeD: fixes a :: "'d"   and b :: "'d" assumes "Le a \<le> Le b" shows "a \<le> b"
A_trusts_OR4: fixes A :: "agent"   and B :: "agent"   and NA :: "msg"   and evs :: "event list"   and B' :: "agent"   and K :: "nat" assumes "Says A B \<lbrace>NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Says B' A \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> otway" shows "\<exists>NB. Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"
bind_le_shift: fixes m :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres"   and m' :: "'c nres" shows "(m \<bind> f \<le> m') = (m \<le> (if nofail m' then SPEC (\<lambda>x. f x \<le> m') else FAIL))"
prev_case: fixes m :: "nat"   and W :: "'a set" assumes "m < n"   and "GRep W"   and "m = fdim W" shows "\<exists>Us. Ball (set Us) IGRep \<and> 0 \<notin> set Us \<and> W = (\<Oplus>U\<leftarrow>Us. U)"
prv_cnj_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp (cnj \<phi> \<chi>) \<psi>)" shows "bprv (imp \<phi> (imp \<chi> \<psi>))"
plane_rel_br: fixes a :: "'a \<Rightarrow> 'c"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>plane_rel = br ((plane_of \<circ>\<circ> map_sctn) a) (\<lambda>x. I (normal x))"
path_disjoint: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "path (xs @ ys)"   and "xs \<noteq> []"   and "x \<in> set xs" shows "x \<notin> set ys"
\<sigma>: shows "\<delta> = \<theta> \<circ>\<^sub>s \<sigma>"
add: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
assms: shows "x \<rightarrow>\<^bsub>H\<^esub> y"
fds_zero_abs_converges: fixes s :: "'a" shows "fds_abs_converges 0 s"
plus_flow_outside: fixes e :: "'a \<times> 'a"   and G :: "('a, 'b) graph_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "e \<notin> \<^bold>E\<^bsub>G\<^esub>" shows "(f \<oplus>\<^bsub>G\<^esub> g) e = 0"
normal_subgroup_axioms: shows "normal_subgroup K G (\<cdot>) \<one>"
rBangTauE: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and S' :: "('a, 'b, 'c) psi" assumes "\<Psi>' \<rhd> !S \<longmapsto> \<tau> \<prec> S'"   and "guarded S" shows "\<exists>T. \<Psi>' \<rhd> S \<parallel> S \<longmapsto> \<tau> \<prec> T \<and> (\<one>, S', T \<parallel> !S) \<in> Rel'"
assms: shows "R \<subseteq> f ` A \<times> f ` A"
mat_det_12_mm_l: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det_12 (M *\<^sub>m\<^sub>m A) (M *\<^sub>m\<^sub>m B) = mat_det M * mat_det_12 A B"
finite_set_mset_mset_set: fixes A :: "'e set" assumes "finite A" shows "set_mset (mset_set A) = A"
sum_zero_eq_neg: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<oplus> y = \<zero>" shows "x = \<ominus> y"
inf_ccSup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'c set"   and a :: "'c" assumes "class.countable_complete_distrib_lattice Inf Sup inf less_eq less sup bot top"   and "countable B" shows "inf a (Sup B) = Sup (inf a ` B)"
step_z_dbm_complete: fixes v :: "'e \<Rightarrow> nat"   and n :: "nat"   and A :: "('g \<times> ('e, 'f) cconstraint \<times> 'h \<times> 'e list \<times> 'g) set \<times> ('g \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'g"   and u :: "'e \<Rightarrow> 'f"   and l' :: "'g"   and u' :: "'e \<Rightarrow> 'f"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" assumes "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"   and "u \<in> [D]\<^bsub>v,n\<^esub>" shows "\<exists>D'. A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle> \<and> u' \<in> [D']\<^bsub>v,n\<^esub>"
measurable_completion: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> completion M \<rightarrow>\<^sub>M N"
welldefined_hyp: shows "all_list_constr (fst (snd (reduce_system p (qs, M_mat signs subsets, subsets, signs)))) (length qs)"
simps_1: fixes f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "(try Bottom catch f) = Bottom"
invariant: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow> b" shows "P b"
strong_induct: fixes P :: "'a \<Rightarrow> pi \<Rightarrow> bool"   and z :: "'a"   and pi :: "pi" assumes "\<And>z. P z \<zero>"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z (name1{name2}.pi)"   and "\<And>pi z. (\<And>z. P z pi) \<Longrightarrow> P z (\<tau>.pi)"   and "\<And>name1 name2 pi z. \<lbrakk>name2 \<sharp> z; name2 \<sharp> name1; \<And>z. P z pi\<rbrakk> \<Longrightarrow> P z (name1<name2>.pi)"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z ([name1\<frown>name2]pi)"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z ([name1\<noteq>name2]pi)"   and "\<And>pi1 pi2 z. \<lbrakk>\<And>z. P z pi1; \<And>z. P z pi2\<rbrakk> \<Longrightarrow> P z (pi1 \<oplus> pi2)"   and "\<And>pi1 pi2 z. \<lbrakk>\<And>z. P z pi1; \<And>z. P z pi2\<rbrakk> \<Longrightarrow> P z (pi1 \<parallel> pi2)"   and "\<And>name pi z. \<lbrakk>name \<sharp> z; \<And>z. P z pi\<rbrakk> \<Longrightarrow> P z (<\<nu>name>pi)"   and "\<And>pi z. (\<And>z. P z pi) \<Longrightarrow> P z (!pi)" shows "P z pi"
sorted_mid_iff2: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "sorted (x # xs @ y # ys) = (sorted (x # xs) \<and> x < y \<and> sorted (xs @ [y]) \<and> sorted (y # ys))"
not_gr: fixes k :: "nat" assumes "m < k" shows "\<not> (k < length vs \<and> vs ! k \<noteq> 0)"
i_append_eq_conv_conj: fixes xs :: "'d list"   and f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" shows "(xs \<frown> f = g) = (xs = g \<Down> length xs \<and> f = g \<Up> length xs)"
eventually_nhds_uniformity: fixes "open" :: "'b set \<Rightarrow> bool"   and uniformity :: "('b \<times> 'b) filter"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "class.uniform_space open uniformity" shows "eventually P (topological_space.nhds open x) = (\<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> P y)"
absorb_comp_converter: fixes d :: "('aa, 'ab) distinguisher"   and conv :: "('aa, 'ab, 'ac, 'ad) converter"   and conv' :: "('ac, 'ad, 'y, 'z) converter" shows "absorb d (conv \<odot> conv') = absorb (absorb d conv) conv'"
sel: fixes a :: "'a" shows "the_gpv (Generative_Probabilistic_Value.Done a) = return_spmf (Pure a)"