hcju/isabelleps huggingface dataset:
insert_closed_def: shows "insert_closed \<equiv> \<lambda>NL N. \<forall>T\<subseteq>Inf ` set NL. \<forall>n\<in>N. strongly_accepts \<F> T (Inf ` set NL \<union> hd NL) \<longrightarrow> T \<lless> {n} \<longrightarrow> strongly_accepts \<F> (insert n T) (Inf ` set NL \<union> hd NL)"
d: shows "d \<in> sphere (0::'a) 1 \<inter> T - h ` (sphere (0::'a) 1 \<inter> S)"
fresh_distinct_fv: fixes S :: "'b set"   and l :: "'a" assumes "atom ` S \<sharp>* l" shows "S \<inter> fv l = {}"
reals_Archimedean: fixes x :: "'c" assumes "(0::'c) < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
insert: fixes a :: "'a"   and A :: "'a set" shows "Lcm\<^sub>f\<^sub>i\<^sub>n (insert a A) = lcm a (Lcm\<^sub>f\<^sub>i\<^sub>n A)"
conseqPre: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q,A"   and "P \<subseteq> P'" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
ValD_strict: fixes f :: "'b \<rightarrow> 'a \<rightarrow> ValD"   and a :: "'b"   and b :: "'a"   and x :: "'b"   and y :: "'a" assumes "f\<cdot>a\<cdot>b = ValTT"   and "f\<cdot>x\<cdot>y = ValFF" shows "f\<cdot>\<bottom>\<cdot>\<bottom> = \<bottom>"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
genPolyTame_subset_genPoly: fixes g' :: "graph"   and i :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" assumes "g' \<in> set (generatePolygonTame i v f g)" shows "g' \<in> set (generatePolygon i v f g)"
sum_sorted_list_of_set_map: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite I" shows "sum_list (map f (list_of I)) = sum f I"
poly_as_sum: fixes p :: "'a poly"   and x :: "'a" shows "poly p x = (\<Sum>i\<le>degree p. x ^ i * coeff p i)"
integrable_restrict_space: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integrable (restrict_space M \<Omega>) f = integrable M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
Res: fixes P :: "pi"   and s :: "(name \<times> name) list"   and Q :: "pi"   and x :: "name" assumes "P[
] \<sim> Q[
]"   and "x \<sharp> s" shows "<\<nu>x>P[
] \<sim> <\<nu>x>Q[
]"
assms: shows "Complete_Partial_Order.chain local.tllist_ord A"
simps_7: fixes v :: "poly" shows "isnpolyh (Neg v) = (\<lambda>k. False)"
vsubst_trans: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and y :: "'var" assumes "good X"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[y2 // y1]_ys = X #[y2 // y]_ys"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "L.ide a"   and "L.seq f a" shows "L.dom f = a"
length_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "List.length (map f xs) = List.length xs"
d_a_galois1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> ad y \<le> d z) = (d x \<le> d z + d y)"
"1_2": shows "(c_, s1_) \<Rightarrow> x_ \<Down> s2_"
rwp_on_Field: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "Field (rwp_on A f) \<subseteq> A"
postdominate_refl: fixes n :: "'node" assumes "valid_node n"   and "\<not> method_exit n" shows "n postdominates n"
rcosets_part_G: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" assumes "Group.group G"   and "subgroup H G" shows "\<Union> (rcosets\<^bsub>G\<^esub> H) = carrier G"
unsorted_poly_rel_list_rel_list_rel_uminus: fixes r :: "(char list list \<times> int) list"   and yc :: "(char list multiset \<times> int) list" assumes "(map (\<lambda>(a, b). (a, - b)) r, yc) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel" shows "(r, map (\<lambda>(a, b). (a, - b)) yc) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"
llength_inf_llist: fixes f :: "nat \<Rightarrow> 'b" shows "llength (inf_llist f) = \<infinity>"
SDG_call_edge: fixes a :: "'edge"   and 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 n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a)" shows "n -p\<rightarrow>\<^bsub>call\<^esub> n'"
integer_of_nat_0: shows "integer_of_nat 0 = 0"
sum_delta_notmem_3: fixes x :: "'a"   and s :: "'a set"   and P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'b" assumes "x \<notin> s" shows "(\<Sum>y\<in>s. if y = x then P y else Q y) = sum Q s"
minus_carrier_mat: fixes B :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'a mat" assumes "B \<in> carrier_mat nr nc" shows "A - B \<in> carrier_mat nr nc"
get_root_node_root_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r root" shows "root |\<in>| object_ptr_kinds h"
assms: shows "\<forall>j\<le>D. str (t + j + (i + k)) = x"
path2_def: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" shows "g \<turnstile> n-ns\<rightarrow>m \<equiv> local.path g ns \<and> n = hd ns \<and> m = last ns"
siso_Atm: fixes atm :: "'atom" shows "siso (Atm atm) = compatAtm atm"
ex_least_nat_le: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n"   and "\<not> P 0" shows "\<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
equiv_negation: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longleftrightarrow>. q) = (A \<turnstile> .\<not> p .\<longleftrightarrow>. .\<not> q)"
setops_list\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) stateful_strand_step list" shows "setops_list\<^sub>s\<^sub>s\<^sub>t S \<equiv> remdups (concat (map setops_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "Right_a.isomorphic a a'" obtains f :: "'a" where "Right_a.in_hom f a a' \<and> Right_a.iso f"
power_0_Suc: fixes n :: "nat" shows "(0::'a) ^ Suc n = (0::'a)"
full_elims_4: fixes n :: "nat"   and l :: "'b tree234"   and p :: "'b"   and m :: "'b tree234"   and q :: "'b"   and m' :: "'b tree234"   and q' :: "'b"   and r :: "'b tree234" assumes "full n \<langle>l, p, m, q, m', q', r\<rangle>" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na m" and "full na m'" and "full na r"
scalar_prod_def: fixes v :: "'a Matrix.vec"   and w :: "'a Matrix.vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $v i * w $v i"
exec_restrict: shows "\<Gamma>|\<^bsub>A\<^esub>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
out: fixes xa :: "'a" shows "d_OUT (\<Squnion> Y) xa = (\<Squnion>f\<in>Y. d_OUT f xa)"
pfxm_prefix_def: fixes prefix_match :: "'b prefix_match" shows "pfxm_prefix prefix_match = (case prefix_match of PrefixMatch x1 x2 \<Rightarrow> x1)"
errMOD_igOpIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igOpIPresIGWls MOD" shows "igOpIPresIGWlsSTR (errMOD MOD)"
distinct_1: fixes x11 :: "linear_poly"   and x12 :: "'a"   and x21 :: "linear_poly"   and x22 :: "'a" shows "LEQ_ns x11 x12 \<noteq> GEQ_ns x21 x22"
preserves_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" shows "fst (B.VV.comp g f) \<star>\<^sub>B snd (B.VV.comp g f) = (fst g \<star>\<^sub>B snd g) \<cdot>\<^sub>B fst f \<star>\<^sub>B snd f"
of_int_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and w :: "int"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus w = ring_1.of_int one plus zero minus z) = (w = z)"
bind_measure_pmf_cong: fixes A :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure"   and B :: "'c \<Rightarrow> 'd measure"   and x :: "'c pmf" assumes "\<And>x. A x \<in> space (subprob_algebra N)"   and "\<And>x. B x \<in> space (subprob_algebra N)"   and "\<And>i. i \<in> set_pmf x \<Longrightarrow> A i = B i" shows "measure_pmf x \<bind> A = measure_pmf x \<bind> B"
lower_dimension_axiom: shows "\<exists>a\<in>unit_disc. \<exists>b\<in>unit_disc. \<exists>c\<in>unit_disc. \<not> poincare_between a b c \<and> \<not> poincare_between b c a \<and> \<not> poincare_between c a b"
conv_complement: fixes x :: "'b" shows "(- x)\<^sup>T = - x\<^sup>T"
sorted_cons: fixes x :: "'b"   and xs :: "'b list" assumes "sorted (x # xs)" shows "sorted xs"
map_pmf_cong: fixes p :: "'a pmf"   and q :: "'a pmf"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "map_pmf f p = map_pmf g q"
cenum_ex_UNIV: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "cEnum = Some (enum, enum_all, enum_ex)" shows "enum_ex P = Bex UNIV P"
simplicial_subdivision_0: fixes p :: "nat" shows "simplicial_subdivision p 0 = 0"
elims: fixes x :: "'b tree"   and xa :: "'b tree"   and y :: "'b tree" assumes "link x xa = y" obtains r :: "nat" and x\<^sub>1 :: "'b" and ts\<^sub>1 :: "'b tree list" and r' :: "nat" and x\<^sub>2 :: "'b" and ts\<^sub>2 :: "'b tree list" where "x = Node r x\<^sub>1 ts\<^sub>1" and "xa = Node r' x\<^sub>2 ts\<^sub>2" and "y = (let t\<^sub>2 = Node r' x\<^sub>2 ts\<^sub>2; t\<^sub>1 = Node r x\<^sub>1 ts\<^sub>1 in if x\<^sub>1 \<le> x\<^sub>2 then Node (r + 1) x\<^sub>1 (t\<^sub>2 # ts\<^sub>1) else Node (r + 1) x\<^sub>2 (t\<^sub>1 # ts\<^sub>2))"
assms: shows "compile_env_entry \<O> \<A> (f, fd1) = Ok (f, fd2)"
ck_trms_sound: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and T :: "'b trm set"   and D :: "'b literal set"   and C :: "'b literal set"   and E :: "'b trm set"   and k :: "inferences" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "T = basic_superposition.get_trms filter_trms D (basic_superposition.dom_trms trm_ord C E) k" shows "T \<subseteq> basic_superposition.dom_trms trm_ord C E"
mm_equiv_def: shows "mm_equiv \<equiv> {(x, y). mm_equivp x y}"
ipd_is_ipd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob" shows "IFC_def.is_ipd prob (IFC_def.ipd prob x) x"
simulation: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and i1 :: "'d"   and s1 :: "'b"   and s2 :: "'c"   and s2' :: "'c" assumes "backward_simulation step1 step2 final1 final2 order match"   and "match i1 s1 s2"   and "step2 s2 s2'" shows "(\<exists>i2 s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match i2 s1' s2') \<or> (\<exists>i2. match i2 s1 s2' \<and> order i2 i1)"
simps_3: fixes P :: "pol"   and x :: "nat"   and Q :: "pol" shows "neg (PX P x Q) = PX (neg P) x (neg Q)"
strip_step: fixes S :: "(char list \<Rightarrow> int) set"   and c :: "(char list \<Rightarrow> int) set acom" shows "strip (step S c) = strip c"
coeff_poly_prod_closed: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'a poly"   and i :: "nat" assumes "\<And>x i. x \<in> X \<Longrightarrow> poly.coeff (f x) i \<in> A" shows "poly.coeff (prod f X) i \<in> A"
min_step_iprev2: fixes x :: "nat"   and y :: "nat"   and I :: "nat set" assumes "x < y"   and "x \<in> I"   and "y \<in> I"   and "\<not> (\<exists>k\<in>I. x < k \<and> k < y)" shows "iprev y I = x"
IH: shows "subst_domain (subst_of L) \<subseteq> set (map fst L)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F H \<tau>\<sigma>.map"
linvf: fixes x :: "'a" assumes "x \<in> carrier K - {\<zero>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<cdot>\<^sub>r x = 1\<^sub>r"
ladder_ix_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_ix L index = (if index = 0 then undefined else deriv_ix (L ! index))"
op_norm_set_proptys_3: fixes A :: "(('b, 'c) vec, 'd) vec" shows "{\<parallel>A *v x\<parallel> |x. \<parallel>x\<parallel> = 1} \<noteq> {}"
mk_poincare_line_cmat_zero_iff: fixes A :: "real"   and B :: "complex" shows "(mk_poincare_line_cmat A B = mat_zero) = (A = 0 \<and> B = 0)"
wt_same_type_typeconf: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e : T" shows "P,E,h \<turnstile> e :\<^bsub>NT\<^esub> T"
absorb1: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" shows "a \<inter># b = a"
ab: fixes x :: "'a" shows "(\<sqinter>xa\<in>range left \<union> range right \<rightarrow>  P) \<le> (right`!`x \<rightarrow> P)"
is_projection_on_eqI: fixes M :: "'a set"   and h :: "'a"   and x :: "'a"   and \<pi> :: "'a \<Rightarrow> 'a" assumes "closed_csubspace M"   and "h - x \<in> orthogonal_complement M"   and "x \<in> M"   and "is_projection_on \<pi> M" shows "\<pi> h = x"
ordering_def: fixes f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "TernaryOrdering.ordering f ord X \<equiv> (\<forall>n. (finite X \<longrightarrow> n < card X) \<longrightarrow> f n \<in> X) \<and> (\<forall>x\<in>X. \<exists>n. (finite X \<longrightarrow> n < card X) \<and> f n = x) \<and> (\<forall>n n' n''. (finite X \<longrightarrow> n'' < card X) \<and> n < n' \<and> n' < n'' \<longrightarrow> ord (f n) (f n') (f n''))"
subst_atm_mset_lists_length: fixes AAs :: "'a multiset list"   and \<sigma>s :: "'s list" shows "length (AAs \<cdot>\<cdot>aml \<sigma>s) = min (length AAs) (length \<sigma>s)"
chain_sup_def: fixes S :: "nat \<Rightarrow> ('c, 'd) labeled_graph" shows "chain_sup S \<equiv> LG (\<Union>i. edges (S i)) (\<Union>i. vertices (S i))"
preserves_arr: fixes f :: "'b" assumes "D.arr f" shows "C.arr (F.right_map f)"
Step1_2_6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S6 rmhist p \<longrightarrow> S1 rmhist p$ \<and> MClkReply memCh crCh cst p \<and> unchanged (e p, r p, m p) \<or> S3 rmhist p$ \<and> MClkRetry memCh crCh cst p \<and> unchanged (e p, r p, m p, rmhist!p)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and list :: "'b list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
arr_Set_arr_ParE: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" obtains "arr_Par \<alpha> T" and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"
A_def: shows "A = {x \<in> space M. l x < \<infinity> \<and> (\<forall>n. l ((T ^^ n) x) = g ((T ^^ n) x))}"
type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap" assumes "type_wf h" shows "DocumentClass.type_wf h"
Re_mult_real3: fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" assumes "is_real z1"   and "is_real z2"   and "is_real z3" shows "Re (z1 * z2 * z3) = Re z1 * Re z2 * Re z3"
factor_composition: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "g \<cdot>| L [\<cdot>] (h \<cdot>| L) = g \<cdot> h \<cdot>| L"
stc_Infinitesimal: fixes x :: "complex star" assumes "x \<in> Infinitesimal" shows "stc x = 0"
homeomorphic_convex_lemma: fixes S :: "'c set"   and T :: "'d set" assumes "convex S"   and "compact S"   and "convex T"   and "compact T"   and "aff_dim S = aff_dim T" shows "S - rel_interior S homeomorphic T - rel_interior T \<and> S homeomorphic T"
FIELD: shows "Field r = {0..<n} \<and> Field p = {0..<Suc n}"
simps: fixes e1 :: "'b"   and a1 :: "'c"   and r1 :: "nat"   and ts1 :: "('b, 'c) BinomialTree list"   and e2 :: "'b"   and a2 :: "'c"   and r2 :: "nat"   and ts2 :: "('b, 'c) BinomialTree list" shows "link (Node e1 a1 r1 ts1) (Node e2 a2 r2 ts2) = (if a1 \<le> a2 then Node e1 a1 (Suc r1) (Node e2 a2 r2 ts2 # ts1) else Node e2 a2 (Suc r2) (Node e1 a1 r1 ts1 # ts2))"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> certified_mail"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> certified_mail; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsfssl X. \<lbrakk>evsfssl \<in> certified_mail; P evsfssl; X \<in> synth (analz (knows Spy evsfssl))\<rbrakk> \<Longrightarrow> P (Notes TTP \<lbrace>Agent Spy, Agent TTP, X\<rbrace> # evsfssl)"   and "\<And>evs1 K q hs cleartext S R m S2TTP. \<lbrakk>evs1 \<in> certified_mail; P evs1; Key K \<notin> used evs1; K \<in> symKeys; Nonce q \<notin> used evs1; hs = Hash \<lbrace>Number cleartext, Nonce q, response S R q, Crypt K (Number m)\<rbrace>; S2TTP = Crypt (pubK TTP) \<lbrace>Agent S, Number BothAuth, Key K, Agent R, hs\<rbrace>\<rbrakk> \<Longrightarrow> P (Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number BothAuth, Number cleartext, Nonce q, S2TTP\<rbrace> # evs1)"   and "\<And>evs2 R S em cleartext q S2TTP hr. \<lbrakk>evs2 \<in> certified_mail; P evs2; Gets R \<lbrace>Agent S, Agent TTP, em, Number BothAuth, Number cleartext, Nonce q, S2TTP\<rbrace> \<in> set evs2; TTP \<noteq> R; hr = Hash \<lbrace>Number cleartext, Nonce q, response S R q, em\<rbrace>\<rbrakk> \<Longrightarrow> P (Notes TTP \<lbrace>Agent R, Agent TTP, S2TTP, Key (RPwd R), hr\<rbrace> # evs2)"   and "\<And>evs3 R S2TTP hr S k hs. \<lbrakk>evs3 \<in> certified_mail; P evs3; Notes TTP \<lbrace>Agent R, Agent TTP, S2TTP, Key (RPwd R), hr\<rbrace> \<in> set evs3; S2TTP = Crypt (pubK TTP) \<lbrace>Agent S, Number BothAuth, Key k, Agent R, hs\<rbrace>; TTP \<noteq> R; hs = hr; k \<in> symKeys\<rbrakk> \<Longrightarrow> P (Notes R \<lbrace>Agent TTP, Agent R, Key k, hr\<rbrace> # Gets S (Crypt (priSK TTP) S2TTP) # Says TTP S (Crypt (priSK TTP) S2TTP) # evs3)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> certified_mail; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)" shows "P x"
ll_def: shows "ll = {p \<in> \<PP>. charslength p = length Doc \<and> terminals p \<in> \<L>}"
simps: fixes a1 :: "trm"   and a2 :: "trm" shows "(a1 \<longlonglongrightarrow>\<^sup>* a2) = ((\<exists>s. a1 = s \<and> a2 = s) \<or> (\<exists>s t u. a1 = s \<and> a2 = u \<and> s \<longlonglongrightarrow> t \<and> t \<longlonglongrightarrow>\<^sup>* u))"
finite_measure_axioms_def: fixes M :: "'a measure" shows "finite_measure_axioms M \<equiv> emeasure M (space M) \<noteq> \<top>"
diff1_3: shows "local.rev_curr g \<in> rst.FINFUNC"
Nothing_def: shows "Nothing \<equiv> maybe_abs\<cdot>(sinl\<cdot>ONE)"
is_extensional: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "\<not> partial_magma.arr A f" shows "\<tau> f = partial_magma.null B"
mcont: shows "mcont lub ord lSup (\<sqsubseteq>) f"
listToFuns_ext: fixes xs :: "'a list"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b list" assumes "set xs = UNIV"   and "distinct xs" shows "(g \<in> set (listToFuns f xs)) = (\<forall>x. g x \<in> set (f x))"
interiorE: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> interior S" obtains T :: "'a set" where "open T" and "x \<in> T" and "T \<subseteq> S"
pos_poly_pCons: fixes a :: "'a"   and p :: "'a poly" shows "pos_poly (pCons a p) = (pos_poly p \<or> p = 0 \<and> (0::'a) < a)"
uniformly_continuous_onD: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and E :: "'c \<times> 'c \<Rightarrow> bool" assumes "uniformly_continuous_on s f"   and "eventually E uniformity" shows "\<forall>\<^sub>F (x, y) in uniformity. x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)"
u_2: shows "u x = return_time {x \<in> S. s x = 0} x"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Match x1 = Match y1) = (x1 = y1)"
pd: shows "\<pi>\<^sub>1 n pd\<rightarrow> \<pi>\<^sub>1 l"
indRelSTEQ_impl_SRel_and_TRel_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelSTEQ SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred SRel (\<lambda>a. hasBarb a SWB)"
alternating_list_alternates: fixes n :: "nat"   and s :: "'b"   and t :: "'b"   and as :: "'b list"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and bs :: "'b list" assumes "alternating_list n s t = as @ [a, b, c] @ bs" shows "a = c"
\<mu>'_1: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j \<le> i" shows "\<mu>' i j = d\<mu> i j"
mintree_exists: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "(bq \<noteq> []) = (SkewBinomialHeapStruc.getMinTree bq \<in> set bq)"
distinct_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (list.set xs))"
routing_ipassmt_wi_sound: fixes tbl :: "'b routing_rule list"   and ea :: "char list"   and eb :: "'b wordinterval"   and k :: "'b word" assumes "valid_prefixes tbl"   and "(ea, eb) \<in> set (routing_ipassmt_wi tbl)"   and "k \<in> wordinterval_to_set eb" shows "ea = output_iface (routing_table_semantics tbl k)"
get_shadow_root_locs_def: shows "get_shadow_root_locs = (\<lambda>element_ptr. {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt)})"
isomorphic_def: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" shows "VV.isomorphic a a' = (\<exists>f. VV.in_hom f a a' \<and> VV.iso f)"
in_keys_mult_scalar_le: fixes v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (p \<odot> q)" shows "v \<preceq>\<^sub>t punit.lt p \<oplus> lt q"
circ_decompose_12: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> * y\<^sup>\<circ> \<le> (x\<^sup>\<circ> * y)\<^sup>\<circ> * x\<^sup>\<circ>"
prime_factorization_unit: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "factorial_semiring.prime_factorization zero times one normalize x = {#}"