hcju/isabelleps huggingface dataset:
measurable_pair_restrict_pmf2: fixes A :: "'b set"   and f :: "'c \<times> 'b \<Rightarrow> 'd"   and M :: "'c measure"   and L :: "'d measure"   and N :: "'b pmf" assumes "countable A"   and "\<And>y. y \<in> A \<Longrightarrow> (\<lambda>x. f (x, y)) \<in> M \<rightarrow>\<^sub>M L" shows "f \<in> M \<Otimes>\<^sub>M restrict_space (measure_pmf N) A \<rightarrow>\<^sub>M L"
natural_transformation_axioms: shows "natural_transformation C C Adj.FG.map Adj.C.map \<epsilon>'"
tab_in_hom_1: shows "\<guillemotleft>tab : src (tab\<^sub>0 r) \<rightarrow> trg r\<guillemotright>"
asubb: shows "fmset G as \<subseteq># fmset G bs"
is_prefix_eq_proper_prefix: fixes a :: "'a list"   and b :: "'a list" shows "is_prefix a b = (a = b \<or> is_proper_prefix a b)"
limI: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow> L" shows "lim X = L"
intros_3: fixes u :: "'e \<Rightarrow> 'd"   and c :: "'e"   and d :: "'d" assumes "u c \<le> d" shows "u \<turnstile> LE c d"
1: shows "{i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} = {i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> i < k} \<union> {i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> k \<le> i}"
carrier_qring1: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "carrier (R /\<^sub>r I) = set_ar_cos R I"
length_Suc_conv: fixes xs :: "'b list"   and n :: "nat" shows "(\<^bold>|xs\<^bold>| = Suc n) = (\<exists>y ys. xs = y # ys \<and> \<^bold>|ys\<^bold>| = n)"
analyzed_in'_def: fixes An :: "'a \<Rightarrow> ('b, 'c) Term.term list \<times> ('b, 'c) Term.term list"   and pu :: "'b \<Rightarrow> bool"   and ar :: "'b \<Rightarrow> nat"   and t :: "'a"   and M :: "('b, 'c) Term.term set" shows "analyzed_in' An pu ar t M \<equiv> case An t of (K, T) \<Rightarrow> (\<forall>k\<in>set K. intruder_synth' pu ar M k) \<longrightarrow> (\<forall>s\<in>set T. intruder_synth' pu ar M s)"
h: shows "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (h t - \<gamma> t) < d / 2)"
apex_def: shows "SPN_rs.apex \<equiv> Maps.dom SPN_rs.leg0"
min_0_1_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq one zero = zero"
fcard_finsert_disjoint: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<notin>| A" shows "fcard (finsert x A) = Suc (fcard A)"
elims_3: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'c list" assumes "\<not> successively x xa" obtains P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xb :: "'c" and y :: "'c" and xs :: "'c list" where "x = P" and "xa = xb # y # xs" and "\<not> (P xb y \<and> successively P (y # xs))"
less_eq_I_def: fixes i :: "I"   and j :: "I" shows "i \<sqsubseteq> j \<equiv> i = j \<or> i \<sqsubset> j"
inf_compl_bot: fixes x :: "'a" shows "inf_class.inf x (- x) = bot_class.bot"
f1: shows "rng_to_rng_of_frac \<in> carrier R \<rightarrow> carrier rec_rng_of_frac"
hvzk_z_r: fixes \<G> :: "'a cyclic_group"   and r1 :: "nat"   and c :: "nat"   and x1 :: "nat" assumes "okamoto \<G>"   and "r1 < order \<G>" shows "r1 = ((r1 + c * x1) mod order \<G> + order \<G> * c * x1 - c * x1) mod order \<G>"
int_nat_equiv: fixes x :: "nat" shows "0 < x \<longrightarrow> nat (int x + - 1) + 1 = x"
enum_ex_unit_def: fixes P :: "unit \<Rightarrow> bool" shows "enum_ex P = P ()"
graph: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "Set_Theory.map \<alpha> S T" shows "\<alpha> \<in> S \<rightarrow>\<^sub>E T"
type_definition_msg: shows "type_definition Rep_Msg Abs_Msg Msg"
mat_vec_as_mat_mat_mult: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "A *\<^sub>v v = col (A * mat_of_cols nc [v]) 0"
residuated_def: fixes f :: "'a \<Rightarrow> 'a" shows "residuated f \<equiv> \<exists>g. residuated_pair f g"
new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr" assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')"   and "ptr \<noteq> new_document_ptr" shows "get ptr h = get ptr h'"
simps_3: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and C' :: "char list"   and Cs :: "char list list"   and a :: "nat"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc (Throwing (C' # Cs) a) frs sh = (None, h, (stk, loc, C, M, pc, Throwing Cs a) # frs, sh(C' \<mapsto> (fst (the (sh C')), Error)))"
map_spmf_cong: fixes p :: "'g spmf"   and q :: "'g spmf"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "map_spmf f p = map_spmf g q"
I_lift_nnf_qe_normal: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "int list" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>\<phi>. \<lbrakk>nqfree \<phi>; Z.normal \<phi>\<rbrakk> \<Longrightarrow> Z.normal (qe \<phi>)"   and "\<And>xs \<phi>. \<lbrakk>Z.normal \<phi>; nqfree \<phi>\<rbrakk> \<Longrightarrow> Z.I (qe \<phi>) xs = (\<exists>x. Z.I \<phi> (x # xs))"   and "Z.normal \<phi>" shows "Z.I (Z.lift_nnf_qe qe \<phi>) xs = Z.I \<phi> xs"
cp_dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S>_at_pre: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(X.boss@pre) \<tau> = (\<lambda>_. X \<tau>.boss@pre) \<tau>"
fds_nth_one_not_Suc_0: fixes n :: "nat" assumes "n \<noteq> Suc 0" shows "fds_nth 1 n = (0::'b)"
Z'_def: fixes n :: "nat" shows "Z' n = emb I (J n) (K n)"
ideal_prod_one: fixes I :: "'a set" assumes "ideal I R" shows "I \<cdot> carrier R = I"
Suc_1: fixes YS :: "(real \<times> real pdevs) list"   and YSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) (YS # YSS)" shows "prove_nonneg prnt i p (slp_of_fas [fa]) YSS"
section_def: fixes f :: "'c" shows "C.section f \<equiv> \<exists>g. C.ide (g \<cdot>\<^sub>C f)"
n: fixes x :: "real" assumes "x \<in> {(real_of_int i' - 1) / 2 ^ (l' + 1)..real_of_int i' / 2 ^ (l' + 1)}" shows "(\<Phi>_n has_real_derivative \<phi> (l', i') x * \<phi> (l, i) x) (at x)"
tauLaw2StatImpRight: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', \<tau>.P, P \<oplus>\<^sub>\<top> \<tau>.P) \<in> Rel" shows "\<Psi> \<rhd> \<tau>.P \<lessapprox><Rel> P \<oplus>\<^sub>\<top> \<tau>.P"
leg1_commutes: shows "\<eta>.cod.leg1 \<cdot> \<eta>.chine = \<eta>.dom.leg1"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" shows "bounded_clinear_axioms f"
A: fixes a :: "nat"   and b :: "nat" shows "indicator {Suc a} (Suc b) = indicator {a} b"
preserves_arr: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "arr (HoVH f)"
eq1: fixes i :: "nat" assumes "i < n - 2" shows "mat_delete (mat_delete (make_mat n k B) 0 k) (k - 1) 0 $$ (i, i) = (1::'a)"
sum_up_index_split: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum f {..m + n} = sum f {..m} + sum f {Suc m..m + n}"
abs_Hahn_Banach: fixes E :: "'c set"   and F :: "'c set"   and f :: "'c \<Rightarrow> real"   and p :: "'c \<Rightarrow> real" assumes "vectorspace E"   and "Subspace.subspace F E"   and "linearform F f"   and "seminorm E p"   and "\<forall>x\<in>F. \<bar>f x\<bar> \<le> p x" shows "\<exists>g. linearform E g \<and> (\<forall>x\<in>F. g x = f x) \<and> (\<forall>x\<in>E. \<bar>g x\<bar> \<le> p x)"
notmatching_notdom: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x (p @ [a]) \<noteq> \<lfloor>a\<rfloor>" shows "x \<notin> dom (Cp a)"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "opq.initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> opq.someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
wellformed_transaction_cases_2: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and l :: "'h strand_label"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "(l, x) \<in> set (transaction_selects T)" shows "(\<exists>t s. x = \<langle>t := s\<rangle>) \<or> (\<exists>t s. x = select\<langle>t,s\<rangle>)"
gb_schema_aux_isGB_init: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and rc :: "nat" assumes "struct_spec sel ap ab compl"   and "compl_conn compl"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ ab gs [] bs data) data"   and "rem_comps_spec (gs @ ab gs [] bs data) (rc, data)" shows "is_Groebner_basis (fst ` set (gb_schema_aux sel ap ab compl gs (rc, data) (ab gs [] bs data) (ap gs [] [] bs data)))"
canonical_cyc_free: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "canonical M n"   and "\<forall>i\<le>n. \<one> \<le> M i i" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
inj_iff_trivial_ker: shows "inj_on (\<lambda>x. finprod G x I) (carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)) = (kernel (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>) (\<lambda>x. finprod G x I) = {\<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>})"
top_ref: fixes x :: "'a" shows "x \<le> \<top>"
not_iAll: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<not> (\<box> t I. P t)) = (\<diamond> t I. \<not> P t)"
sswSts_def: shows "sswSts = (SOME wSts'. full_chain (\<leadsto>\<^sub>w) wSts' \<and> emb wSts wSts' \<and> lhd wSts' = lhd wSts \<and> llast wSts' = llast wSts)"
assms_10: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and e :: "exp" assumes "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S"   and "I (\<Gamma>, Let \<Delta> e, S)" shows "P (\<Gamma>, Let \<Delta> e, S) (\<Delta> @ \<Gamma>, e, S)"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int" shows "bval (Less a\<^sub>1 a\<^sub>2) s = (aval a\<^sub>1 s < aval a\<^sub>2 s)"
order_top_ow_axioms: shows "order_top_ow U (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w) \<top>\<^sub>o\<^sub>w"
total_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "times_top_class.total x" shows "times_top_class.total (x \<squnion> y)"
nextl_app: fixes q :: "hf"   and xs :: "'a list"   and ys :: "'a list" shows "Accessible.nextl q (xs @ ys) = Accessible.nextl (Accessible.nextl q xs) ys"
ind_hom_injec: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "injec\<^bsub>A /\<^sub>r (ker\<^bsub>A,R\<^esub> f),R\<^esub> (f\<degree>\<^bsub>A,R\<^esub>)"
card_UNIV_unit_def: shows "card_UNIV_class.card_UNIV = Phantom(unit) 1"
final_fv: fixes e :: "char list exp" assumes "final e" shows "fv e = {}"
prime_factorization_0: shows "prime_factorization (0::'a) = {#}"
subset_imp_msubset_mset_set: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "finite B" shows "mset_set A \<subseteq># mset_set B"
set_split: fixes L :: "atomUni list"   and eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list" assumes "separateAtoms L = (eq, les, leq, neq)" shows "set L = set (map EqUni eq @ map LessUni les @ map LeqUni leq @ map NeqUni neq)"
less_finite_5_def: fixes x :: "finite_5"   and y :: "finite_5" shows "(x < y) = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> False | (a\<^sub>1, _) \<Rightarrow> True | (a\<^sub>2, a\<^sub>3) \<Rightarrow> True | (a\<^sub>2, a\<^sub>5) \<Rightarrow> True | (a\<^sub>2, _) \<Rightarrow> False | (a\<^sub>5, x) \<Rightarrow> False | (_, a\<^sub>5) \<Rightarrow> True | (_, _) \<Rightarrow> False)"
integral_combine_division_bottomup: fixes \<D> :: "'c set set"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<D> division_of S"   and "\<And>k. k \<in> \<D> \<Longrightarrow> f integrable_on k" shows "integral S f = (\<Sum>i\<in>\<D>. integral i f)"
inter_sgsTr1: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g"   and "i < r * s" shows "G \<guillemotright> f (i div s) \<inter> g (s - Suc 0) "
min_term_def: shows "min_term \<equiv> term_of_pair (0::'a, the_min)"
allUses_def_ass: fixes v :: "'a \<times> 'b \<times> Def"   and g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b" assumes "v \<in> gen_wf.braun_ssa.allUses g n"   and "n \<in> set (gen_wf_\<alpha>n g)" shows "gen_wf.braun_ssa.defAss g n v"
Inv_Ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.Inv C a = a"
WT_obsf_converter: fixes \<I> :: "('m, 'n) \<I>"   and \<I>' :: "('o, 'p) \<I>"   and conv :: "('m, 'n, 'o, 'p) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>" shows "exception_\<I> \<I>, exception_\<I> \<I>' \<turnstile>\<^sub>C obsf_converter conv \<surd>"
unset_bit_nonnegative_int_iff: fixes n :: "nat"   and k :: "int" shows "(0 \<le> unset_bit n k) = (0 \<le> k)"
gt_halfspace_def: fixes sctn :: "'e sctn"   and x :: "'e" shows "gt_halfspace sctn x = (pstn sctn < x \<bullet> normal sctn)"
smult_prod_extract2: fixes a :: "'b"   and A :: "'b tensor"   and B :: "'b tensor" shows "a \<cdot> (A \<otimes> B) = A \<otimes> (a \<cdot> B)"
int_to_nat_raw_def: shows "int_to_nat_raw \<equiv> \<lambda>(x, y). x - y"
inject: fixes x :: "heap \<Rightarrow> ('a \<times> heap) option"   and ya :: "heap \<Rightarrow> ('a \<times> heap) option" shows "(Heap x = Heap ya) = (x = ya)"
new_def: fixes n :: "nat"   and x :: "'a" shows "Array.new n x = Heap_Monad.heap (Array.alloc (replicate n x))"
var_def: fixes i :: "'b" shows "var i = Abs_formula {A. i \<in> A}"
authKeys_def: fixes evs :: "event list" shows "authKeys evs = {authK. \<exists>A Peer Ta. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Peer, Ta\<rbrace>, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key authK, Ta\<rbrace>\<rbrace> \<in> set evs}"
cases: fixes x :: "nat \<times> 'a mat" obtains j :: "nat" and A :: "'a mat" where "x = (j, A)"
S_2: shows "normalize n = (\<Prod>p\<in>S. p ^ f p)"
is_operator_applicable_in_def: fixes s :: "'c \<Rightarrow> bool option"   and op :: "'c strips_operator" shows "STRIPS_Representation.is_operator_applicable_in s op \<equiv> Let (strips_operator.precondition_of op) (list_all (\<lambda>v. s v = Some True))"
vimageI: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'a"   and B :: "'a set" assumes "f a = b"   and "b \<in> B" shows "a \<in> f -` B"
2: shows "True \<notin> set ((harry_step ^^ y) xs)"
set_alternating_list1: fixes n :: "nat"   and s :: "'b"   and t :: "'b" assumes "order.greater_eq n 1" shows "s \<in> set (alternating_list n s t)"
fun_upd_triv: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g" shows "f(x := f x) = f"
e: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> e) = (p = q)"
frontier_of_topspace: fixes X :: "'b topology" shows "X frontier_of topspace X = {}"
If_not: fixes b :: "bool"   and x :: "'a"   and y :: "'a" shows "(if \<not> b then x else y) = (if b then y else x)"
instrs: shows "instrs sb @ is\<^sub>s\<^sub>b = is @ prog_instrs sb"
Sup_to_Inf_var: fixes X :: "'a set" shows "\<Squnion> X = \<partial> (\<Sqinter> (\<partial> ` X))"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and LV :: "char list set" shows "qdeps (C\<^sub>1;; C\<^sub>2) LV = qdeps C\<^sub>1 (qdeps C\<^sub>2 LV)"
rep_eq: fixes x :: "\<Pi>\<^sub>3" shows "d\<^sub>3 x = Some (eval\<Pi>\<^sub>3 x)"
HInfinite_hcmod_iff: fixes z :: "complex star" shows "(z \<in> HInfinite) = (hcmod z \<in> HInfinite)"
cexpr_comp_def: fixes b :: "cexpr"   and a :: "cexpr" shows "b \<circ>\<^sub>c a \<equiv> cexpr_comp_aux 0 a b"
js_no: fixes s :: "'a list \<Rightarrow> 'a" shows "induced_play (joint_strategy s defensive_strategy_Odd) p \<notin> A"
differentiable_at_snd: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and x :: "'b"   and X :: "'b set" assumes "f differentiable at x within X" shows "(\<lambda>x. snd (f x)) differentiable at x within X"
bex_triv_one_point1: fixes A :: "'c set"   and a :: "'c" shows "(\<exists>x\<in>A. x = a) = (a \<in> A)"
merge_length_less2: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs" shows "length vs \<le> length as"
dual_upset_set: shows "ord.upset_set (\<lambda>x y. y \<le> x) = \<Down>"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "category C" shows "identity_functor C"
connectedI: fixes xs :: "Vertices list"   and v :: "Vertices"   and w :: "Vertices" assumes "walk xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "connected v w"
funpow_dist_0: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x = y" shows "funpow_dist f x y = 0"
can_syncE: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>1" obtains taa :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" and x' :: "'x1" and m' :: "'m1" where "r1.r_syntax t x m taa x' m'" and "LT = collect_waits taa"
converse_SubobjsR_Rep: 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 C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs\<^sub>R P C Cs"   and "P \<turnstile> last Cs \<prec>\<^sub>R C'"   and "wf_prog wf_md P" shows "Subobjs\<^sub>R P C (Cs @ [C'])"
djf_Or: fixes bs :: "real list"   and f :: "'a \<Rightarrow> fm"   and p :: "'a"   and q :: "fm" shows "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
make\<o>_cases: fixes x :: "\<o>" obtains y :: "j \<Rightarrow> i \<Rightarrow> bool" where "x = make\<o> y" and "y \<in> UNIV"
sinterleave_snth_2: fixes n :: "nat"   and s1 :: "'d stream"   and s2 :: "'d stream" assumes "odd n" shows "sinterleave s1 s2 !! n = s2 !! (n div 2)"
path_edge: fixes l1' :: "'vtx"   and lbl :: "'lbl"   and l2' :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and l1 :: "'vtx"   and l2 :: "'vtx" assumes "(l1', lbl, l2') \<in> set xs"   and "path l1 l2 xs" shows "lbl \<in>\<^sub>A weights l1' l2'"
eexp_mono2: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y" shows "eexp x \<le> eexp y"
f9: shows "s \<otimes> s'' \<otimes> (s' \<otimes> s'') = s \<otimes> s' \<otimes> (s'' \<otimes> s'')"
pwd_spy_id_pubk_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "let (s, S, A, B, C, D) = pwd_spy_id_pubk n; SK = (Some S, {A, B}, {C, D}) in pwd_spy_id_pubk_state n (S, A, B, C, D) \<subseteq> s \<and> {SesKey SK} \<in> key_sets (Pwd n) (crypts (Log -` spied s))"
pfun_subset_eq_least: fixes f :: "('a, 'b) pfun" shows "{}\<^sub>p \<subseteq>\<^sub>p f"
summable_mult2: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "summable f" shows "summable (\<lambda>n. f n * c)"
wens_Join_subset: fixes F :: "'b program"   and G :: "'b program"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens (F \<squnion> G) act B \<subseteq> wens F act B"
alphaAbs_weakestEx_imp_strongestAll: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X'" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X'"
labels_not_empty: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "labels \<Gamma> p \<noteq> {}"
length_induced_play: fixes A :: "'b llist set"   and N :: "enat"   and s :: "'c list \<Rightarrow> 'c"   and p :: "'c list" assumes "GSgame A N" shows "llength (GSgame.induced_play N s p) = 2 * N"
Rep_uint8'_def: shows "Rep_uint8' = Rep_uint8"