hcju/isabelleps huggingface dataset:
valid_Act: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and x :: "('b, unit, ('act, 'pred) S_action) formula" shows "P \<Turnstile>\<^sub>S Formula.Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Formula.Act \<alpha> x = Formula.Act \<alpha>'' x' \<and> P \<rightarrow>\<^sub>S \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile>\<^sub>S x')"
new_Addr_SomeD: fixes h :: "heap"   and a :: "nat" assumes "new_Addr h = \<lfloor>a\<rfloor>" shows "h a = None"
SIod_less: fixes T :: "'c set"   and D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<prec>\<^bsub>SIod D T\<^esub> b = a \<prec>\<^bsub>D\<^esub> b"
space_K: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "space (K x) = space S"
rows_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "rows (Finite_Cartesian_Product.transpose A) = columns A"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "Span.dom (SPN f) = SPN (local.dom f)"
postList_preList: fixes C :: "acom"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "postList (map (\<lambda>(x, e). (x, preT C e)) upds) l s = preList upds C l s"
assms_2: shows "Complete_Partial_Order.chain (\<le>) (f ` Y)"
ex_y': fixes x :: "'a" assumes "x \<in># X + Ya" shows "\<exists>y. y \<in># Y - Ya + {#z#} \<and> gt y x"
to_bool_neq_0: fixes x :: "'b word" shows "to_bool x = (x \<noteq> 0)"
app_Var_NF: fixes t :: "dB"   and i :: "nat" assumes "NF t" shows "\<exists>t'. t \<degree> dB.Var i \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
Cons_eq_map_conv: fixes x :: "'a"   and xs :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and ys :: "'b list" shows "(x # xs = map f ys) = (\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs)"
c_rec: shows "c_rec \<le> (\<Sum>jj = Suc j..<i. 2 * n + 2 + 4 * jj + 1)"
frange_mono: fixes r :: "V"   and s :: "V" assumes "r \<subseteq>\<^sub>\<circ> s" shows "\<R>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> \<R>\<^sub>\<bullet> s"
coset_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier H'" shows "M #>\<^bsub>H'\<^esub> \<one>\<^bsub>H'\<^esub> = M"
assms_2: shows "is_serial_solution_for_problem \<Psi> \<psi>"
reduce_le_plus_consts': fixes n2 :: "nat"   and n1 :: "nat"   and x :: "nat"   and y :: "nat" assumes "n2 \<le> n1"   and "x + n1 \<le> y + n2" shows "x + (n1 - n2) \<le> y"
a2: shows "Y \<in> set (ftv (lookup \<theta> X))"
intro: fixes p :: "int" assumes "prime p" shows "poly_mod_prime p"
listset_empty: fixes XS :: "'b set list" shows "(listset XS = {}) = (\<not> list_all (\<lambda>A. A \<noteq> {}) XS)"
ord_spmf_eq_leD: fixes p :: "'a spmf"   and q :: "'a spmf"   and x :: "'a" assumes "ord_spmf (=) p q" shows "spmf p x \<le> spmf q x"
diff_diff_right: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "c - (b - a) = c + a - b"
circline_set_x_axis: shows "circline_set x_axis = of_complex ` {x. is_real x} \<union> {\<infinity>\<^sub>h}"
le_def: fixes r\<^sub>A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "Product.le r\<^sub>A r\<^sub>B = (\<lambda>(a\<^sub>1, b\<^sub>1) (a\<^sub>2, b\<^sub>2). a\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2)"
swap_lens_snd: shows "snd\<^sub>L ;\<^sub>L swap\<^sub>L = fst\<^sub>L"
C_is: shows "C = {(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}"
remdupsAppendMultiSet: fixes a :: "'b list"   and b :: "'b list" shows "mset (remdups (a @ b)) = mset (remdups a @ remdups (list_diff b a))"
units_eq: fixes e1 :: "'a"   and e2 :: "'a" assumes "e1 \<in> E"   and "e2 \<in> E"   and "D e1 e2" shows "e1 = e2"
pcr_cr_eq: shows "transaction_merkle.pcr_transaction = transaction_merkle.cr_transaction"
measurable_fst: fixes M1 :: "'b measure"   and M2 :: "'c measure" shows "fst \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M1"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid left_forkee)]), left_forkee \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
assms: shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>"
open_map_product_projection: fixes i :: "'c"   and I :: "'c set"   and Y :: "'c \<Rightarrow> 'd topology" assumes "i \<in> I" shows "open_map (product_topology Y I) (Y i) (\<lambda>f. f i)"
assms_3: shows "tfr\<^sub>s\<^sub>t \<A>"
tLength_Suc: fixes t :: "'b Trace"   and n :: "nat" assumes "tLength t = Suc n" shows "\<exists>s t'. t = t' \<leadsto> s \<and> tLength t' = n"
representsD1: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and r :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" assumes "pmdl.represents B r x" shows "keys r \<subseteq> B"
perm_vars_lt: fixes n :: "nat"   and vs :: "nat list" assumes "perm_vars n vs" shows "\<forall>i\<in>set vs. i < n"
dominates_antitrans: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and g :: "'a"   and n\<^sub>1 :: "'b"   and m :: "'b"   and n\<^sub>2 :: "'b" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry"   and "invar g"   and "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n\<^sub>1 m"   and "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n\<^sub>2 m" obtains "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n\<^sub>1 n\<^sub>2"   | "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n\<^sub>2 n\<^sub>1"
map_vec_def: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c vec" shows "map_vec f v \<equiv> vec (dim_vec v) (\<lambda>i. f (v $ i))"
ChamberSubcomplex_def: fixes Y :: "'a set set" shows "folding.ChamberSubcomplex Y \<equiv> order.greater_eq X Y \<and> ChamberComplex Y \<and> (\<forall>C. SimplicialComplex.maxsimp Y C \<longrightarrow> folding.chamber C)"
assms: shows "\<forall>i\<in>I. |A i| =o |B i|"
assms_2: shows "(E1, E2) \<in> mult1 r"
summable_Leibniz_1: fixes a :: "nat \<Rightarrow> real" assumes "a \<longlonglongrightarrow> 0"   and "monoseq a" shows "summable (\<lambda>n. (- 1) ^ n * a n)"
1: fixes T :: "'b set" assumes "T \<in> sets borel" shows "{x \<in> S. f x \<in> T} \<in> sets lebesgue"
app_invimageD2: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A"
Axiom_def: shows "Axiom \<equiv> Extra_ax \<union> Sent \<union> Equality_ax \<union> HF_ax \<union> Special_ax \<union> Induction_ax"
elims: fixes x :: "nat"   and y :: "'a fract" assumes "s.h x = y" obtains i :: "nat" where "x = i" and "y = (if i \<le> 1 then 1 else if i = 2 then lead_coeff (local.F 2) ^ (degree (local.F 1) - degree (local.F (Suc 1))) else lead_coeff (local.F i) ^ (degree (local.F (i - 1)) - degree (local.F (Suc (i - 1)))) / s.h (i - 1) ^ (degree (local.F (i - 1)) - degree (local.F (Suc (i - 1))) - 1))"
assms: shows "(x, s') \<in> results_gpv \<I>' (inline_aux callee y)"
preserves_seq: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "D.seq (F g) (F f)"
first_id_def: fixes x :: "'a" shows "first_id x = x"
face_cycle_succ_conv: shows "face_cycle_succ = lists_fc_succ xss"
simps_10: fixes v1 :: "'a"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (V v1) (Times v va) = True"
89: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - y \<sqinter> x = x"
distance_def: fixes v :: "'a"   and w :: "'a" shows "distance v w \<equiv> Min {length xs |xs. v \<leadsto>xs\<leadsto> w}"
subseq_drop_many: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "subseq xs ys" shows "subseq xs (zs @ ys)"
induct: fixes x1 :: "(nat \<times> bval) list"   and x2 :: "exp"   and x3 :: "bval"   and P :: "(nat \<times> bval) list \<Rightarrow> exp \<Rightarrow> bval \<Rightarrow> bool" assumes "x1 \<turnstile> x2 \<Down> x3"   and "\<And>\<rho> n. P \<rho> (ENat n) (BNat n)"   and "\<And>\<rho> x v. lookup \<rho> x = Some v \<Longrightarrow> P \<rho> (EVar x) v"   and "\<And>\<rho> x e. P \<rho> (ELam x e) (BClos x e \<rho>)"   and "\<And>\<rho> e1 x e \<rho>' e2 arg v. \<lbrakk>\<rho> \<turnstile> e1 \<Down> BClos x e \<rho>'; P \<rho> e1 (BClos x e \<rho>'); \<rho> \<turnstile> e2 \<Down> arg; P \<rho> e2 arg; (x, arg) # \<rho>' \<turnstile> e \<Down> v; P ((x, arg) # \<rho>') e v\<rbrakk> \<Longrightarrow> P \<rho> (EApp e1 e2) v"   and "\<And>\<rho> e1 n1 e2 n2 n3 f. \<lbrakk>\<rho> \<turnstile> e1 \<Down> BNat n1; P \<rho> e1 (BNat n1); \<rho> \<turnstile> e2 \<Down> BNat n2; P \<rho> e2 (BNat n2); n3 = f n1 n2\<rbrakk> \<Longrightarrow> P \<rho> (EPrim f e1 e2) (BNat n3)"   and "\<And>\<rho> e1 e3 v3 e2. \<lbrakk>\<rho> \<turnstile> e1 \<Down> BNat 0; P \<rho> e1 (BNat 0); \<rho> \<turnstile> e3 \<Down> v3; P \<rho> e3 v3\<rbrakk> \<Longrightarrow> P \<rho> (EIf e1 e2 e3) v3"   and "\<And>\<rho> e1 n e2 v2 e3. \<lbrakk>\<rho> \<turnstile> e1 \<Down> BNat n; P \<rho> e1 (BNat n); n \<noteq> 0; \<rho> \<turnstile> e2 \<Down> v2; P \<rho> e2 v2\<rbrakk> \<Longrightarrow> P \<rho> (EIf e1 e2 e3) v2" shows "P x1 x2 x3"
arl_last_def: shows "arl_last \<equiv> \<lambda>(a, n). Array.nth a (n - 1)"
treap_of_bst_of_list': fixes ys :: "('c \<times> 'd) list"   and p :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and xs' :: "'c list" assumes "ys = map (\<lambda>x. (x, p x)) xs"   and "inj_on p (set xs)"   and "xs' = sort_key p xs" shows "map_tree fst (treap_of (set ys)) = bst_of_list xs'"
BC: shows "\<nu>.dom.leg0 \<cdot> doms.Prj\<^sub>1\<^sub>0 = \<pi>.dom.leg1 \<cdot> doms.Prj\<^sub>0\<^sub>0"
functor_axioms: shows "functor A C (L \<circ> F)"
image_add_atLeast: fixes k :: "nat"   and i :: "nat" shows "(\<lambda>n. n + k) ` {i..} = {i + k..}"
simps_1: fixes v :: "variable"   and S :: "String.literal set" shows "tvsT (Tv v S) = {(v, S)}"
is_comm_monoid_cancel: shows "comm_monoid_cancel (Ring_Divisibility.mult_of R)"
nat_distrib_4: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k * (m - n) = k * m - k * n"
injec_inj_on: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup F"   and "aGroup G"   and "injec\<^bsub>F,G\<^esub> f" shows "inj_on f (carrier F)"
MOST_iff_finiteNeg: fixes P :: "'c \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
tstp_le_def: fixes tstp :: "nat + nat"   and ts :: "nat"   and tp :: "nat" shows "tstp_le tstp ts tp = (case tstp of Inl ts' \<Rightarrow> ts' \<le> ts | Inr tp' \<Rightarrow> tp' \<le> tp)"
subset_closed_segment: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "(closed_segment a b \<subseteq> closed_segment c d) = (a \<in> closed_segment c d \<and> b \<in> closed_segment c d)"
map_transform_Nil: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "map_transform t ae [] = []"
insert_def: shows "DList_Set.insert \<equiv> map_fun id (map_fun list_of_dlist Abs_dlist) (equal_base.list_insert ceq')"
prod_emb_Pi: fixes X :: "'c \<Rightarrow> 'd set"   and J :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and K :: "'c set" assumes "X \<in> (\<Pi> j\<in>J. sets (M j))"   and "J \<subseteq> K" shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
ltln_weak_to_strong_3: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi>"   and "\<not> w \<Turnstile>\<^sub>n G\<^sub>n \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi>"
eq: shows "(\<lambda>(t, s, \<omega>). trace_in ss t s \<omega>) = lfp (\<lambda>F x. case x of (t, s, (t', s') ## \<omega>) \<Rightarrow> if t < t' then s \<in> ss else F (t, s', \<omega>))"
set_pmf_subset_singleton: fixes p :: "'b pmf"   and x :: "'b" shows "(set_pmf p \<subseteq> {x}) = (p = return_pmf x)"
fpxs_of_fls_compose_power: fixes f :: "'b fls"   and d :: "nat" shows "fpxs_of_fls (fls_compose_power f d) = fpxs_compose_power (fpxs_of_fls f) (rat_of_nat d)"
known_ptr_def: shows "NodeClass.known_ptr \<equiv> l_known_ptr\<^sub>N\<^sub>o\<^sub>d\<^sub>e.a_known_ptr"
at_bij: fixes pi :: "('a \<times> 'a) list"   and x :: "'a"   and y :: "'a" assumes "at TYPE('a)" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
L2_set_infinite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "infinite A" shows "L2_set f A = 0"
euler_genus_def: shows "GM.euler_genus \<equiv> (int (2 * card G.sccs) - int (card G.isolated_verts) - GM.euler_char) div 2"
assms_2: shows "topspace T1 \<subseteq> g -` A"
rewrite_negated_dst_ports_not_has_disc_negated: fixes m :: "'c common_primitive match_expr" assumes "normalized_nnf_match m" shows "\<not> has_disc_negated is_Dst_Ports False (rewrite_negated_dst_ports m)"
Mojmir_rabin_smallest_accepting_rank: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F" shows "mojmir_def.smallest_accepting_rank \<Sigma> \<delta> q\<^sub>0 w F = mojmir_to_rabin.smallest_accepting_rank\<^sub>\<R> \<Sigma> \<delta> q\<^sub>0 w F"
add_comm: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
P_def: shows "P = (\<lambda>x. {p. prime p \<and> real p \<le> x})"
sets_count_space: fixes \<Omega> :: "'d set" shows "sets (count_space \<Omega>) = Pow \<Omega>"
6: shows "g_E G' `` {v} \<subseteq> E `` {v}"
DB_base: shows "basevars (inp, mid, out, q1, q2)"
assms_1: fixes x :: "'a vec" assumes "x \<in> set xs" shows "x \<in> carrier_vec n"
pair_measure_closed: fixes A :: "'c measure"   and B :: "'d measure" shows "{a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)"
iprev_inext_fin: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<noteq> Max I" shows "iprev (inext n I) I = n"
extremum_strict: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "ordering_top less_eq less top" shows "\<not> less top a"
Nullable_def: fixes R :: "'b rexp set" shows "Nullable R \<equiv> \<exists>r\<in>R. nullable r"
nb\<^sub>1: shows "(\<Phi>\<inverse> \<Pi> \<A> t) ! k = decode_plan' \<Pi> \<A> k"
instInpP1: fixes \<sigma> :: "'fmla"   and \<tau> :: "'fmla"   and n :: "nat" assumes "\<sigma> \<in> ptrm (Suc 0)"   and "\<tau> \<in> ptrm n" shows "instInpP \<sigma> n \<tau> \<in> ptrm n"
indep: shows "\<forall>s\<^sub>1 s\<^sub>2. Tr\<^sub>1 s\<^sub>1 \<inter> Tr\<^sub>2 s\<^sub>2 = {}"
\<beta>_boundedness_ge: fixes m :: "int"   and x :: "'a" assumes "m \<le> int (k x)"   and "x \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. real_of_int m \<le> u x}"
reflexive_complement_irreflexive: fixes x :: "'a" assumes "reflexive x" shows "irreflexive (- x)"
Gcd_Lcm: fixes A :: "'b set" shows "normalization_euclidean_semiring_class.Gcd A = normalization_euclidean_semiring_class.Lcm {b. \<forall>a\<in>A. b dvd a}"
cl_subset_in_lattice: fixes L :: "'b set set"   and r :: "'b set" assumes "cl L r \<subseteq> r"   and "lattice L" shows "r \<in> L"
subsumption2: fixes Q :: "'a upred"   and P :: "'a upred" assumes "`Q \<Rightarrow> P`" shows "(P \<or> Q) = P"
wf_mdecl_def: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, Ts, T, m). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> (case m of None \<Rightarrow> C\<bullet>M(Ts) :: T | \<lfloor>mb\<rfloor> \<Rightarrow> wf_md P C (M, Ts, T, mb))"
Prefixes_def: fixes p :: "'b list" shows "Prefixes p = {q. is_prefix q p}"
args_to_set_def: fixes x :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list \<times> (('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list \<times> ((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "args_to_set x = fst ` (set (fst x) \<union> set (fst (snd x)) \<union> fst ` set (snd (snd x)) \<union> snd ` set (snd (snd x)))"
integral_return: fixes x :: "'b"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>L (return M x) g = g x"
in_set_list_format: fixes e :: "'g"   and l :: "'g list" assumes "e \<in> set l" obtains l1 :: "'g list" and l2 :: "'g list" where "l = l1 @ e # l2"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and x3 :: "('fun, 'var) extstrand_step list"   and P :: "('fun, 'var) Term.term set \<Rightarrow> ('var \<Rightarrow> ('fun, 'var) Term.term) \<Rightarrow> ('fun, 'var) extstrand_step list \<Rightarrow> bool" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d x1 x2 x3"   and "\<And>M\<^sub>0 \<I>. P M\<^sub>0 \<I> []"   and "\<And>M\<^sub>0 \<I> S t. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"   and "\<And>M\<^sub>0 \<I> S t. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"   and "\<And>M\<^sub>0 \<I> S t t' a. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; Unifier \<I> t t'\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>M\<^sub>0 \<I> S X F. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ineq_model \<I> X F\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"   and "\<And>M\<^sub>0 \<I> S t K M. \<lbrakk>sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S; P M\<^sub>0 \<I> S; ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>; Ana t = (K, M); \<And>k. k \<in> set K \<Longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> k \<cdot> \<I>\<rbrakk> \<Longrightarrow> P M\<^sub>0 \<I> (S @ [Decomp t])" shows "P x1 x2 x3"
assms_4: shows "length xvec2 = residualLength cRs"
Decrypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key (invKey K) \<in> analz H" shows "X \<in> analz H"
axioms_5: 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 \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "natural_transformation A B F G \<tau>"
top: shows "\<Omega> \<in> sigma_sets \<Omega> E"
rep_eq: fixes x :: "circline_mat" shows "pos_oriented_clmat x = pos_oriented_cmat (Rep_circline_mat x)"
Eq: fixes t :: "tm"   and u :: "tm"   and i :: "name"   and x :: "tm" shows "(t EQ u)(i::=x) = subst i x t EQ subst i x u"
quotient_of_submodules_inc_0: fixes P :: "'a set"   and Q :: "'a set" assumes "submodule R M P"   and "submodule R M Q" shows "\<zero>\<^bsub>R\<^esub> \<in> P \<^bsub>R\<ddagger>M\<^esub> Q"
absurd_PL: fixes A :: "'a set"   and p :: "'a" assumes "A \<turnstile> .\<not> (.\<not> p)" shows "A \<turnstile> p"
ende: shows "T\<^sub>p [x, y] d (OPT2 d [x, y]) = 0"
c1: assumes "0 \<le> b" shows "\<exists>w. \<forall>y<w. b * y < y\<^sup>2"
sasp_filter_empty_action: fixes s :: "'a \<Rightarrow> 'b option"   and \<pi>s :: "('a, 'b) sas_plus_operator list" shows "execute_serial_plan_sas_plus s (rem_noops \<pi>s) = execute_serial_plan_sas_plus s \<pi>s"
ltln_Release_to_WeakUntil: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> W\<^sub>n (\<phi> and\<^sub>n \<psi>)"
fps_integral0_fps_const: fixes c :: "'a" shows "fps_integral0 (fps_const c) = fps_const c * fps_X"
mult2_alt_local: fixes X :: "'a multiset"   and Y :: "'a multiset"   and b :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> mult2_alt b ns s" shows "(X, Y) \<in> mult2_alt b (ns \<inter> set_mset X \<times> set_mset Y) (s \<inter> set_mset X \<times> set_mset Y)"
fminus_idemp: fixes A :: "'a fset"   and B :: "'a fset" shows "A |-| B |-| B = A |-| B"
word_pred_m1: fixes a :: "'b word" shows "word_pred a = a - 1"
arg_cong: fixes x :: "'e"   and y :: "'e"   and f :: "'e \<Rightarrow> 'f" assumes "x = y" shows "f x = f y"
semiconfluentpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r\<^sup>*\<^sup>* x z\<rbrakk> \<Longrightarrow> \<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u" shows "semiconfluentp r"
MRList3: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and xa :: "((int \<times> int) set set, int) Combinators"   and a :: "((int \<times> int) set set, int) Combinators list"   and b :: "((int \<times> int) set set, int) Combinators"   and xs :: "((int \<times> int) set set, int) Combinators list" assumes "x \<notin> dom (C xa)" shows "applied_rule_rev C x (a @ b # xs @ [xa]) = applied_rule_rev C x (a @ b # xs)"
gen_subalgebra_def: fixes M :: "'c measure"   and G :: "'c set set" shows "gen_subalgebra M G = sigma (space M) G"
simps_17: fixes C' :: "char list"   and e :: "'a exp" shows "not_init C' (throw e) = not_init C' e"
entail_sat: fixes \<Gamma> :: "'b formula set" shows "\<Gamma> \<TTurnstile> \<bottom> = (\<not> sat \<Gamma>)"
create_gba__incoming_exists: shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. \<forall>nd\<in>g_V \<A>. incoming nd \<subseteq> insert expand_init (name ` g_V \<A>))"
iD_isD: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []"   and "x \<notin> dom (Cp (list2FWpolicy p))" shows "Cp (DenyAll \<oplus> list2FWpolicy (insertDenies p)) x = Cp DenyAll x"
a: shows "a = T (norm a *\<^sub>R axis k 1)"
test_not: fixes p :: "'a" shows "\<iota> (- p) = - (\<iota> p)"
SN: shows "SN_rel_ext P' Pw' R' Rw' M"
map_update: fixes m :: "'a"   and a :: "'v"   and b :: "nat" assumes "M_invar m" shows "M_lookup (M_update a b m) = M_lookup m(a \<mapsto> b)"
arrow_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" shows "arrow_of_spans C \<mu> \<equiv> (category C \<and> span_in_category C (Dom \<mu>)) \<and> span_in_category C (Cod \<mu>) \<and> arrow_of_spans_axioms C \<mu>"
weakSimReflexive: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "{(\<Psi>, P, P) |\<Psi> P. True} \<subseteq> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> P"
absorb_iff2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
rep_eq: fixes x :: "uint16"   and xa :: "nat"   and xb :: "bool" shows "Rep_uint16 (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (Rep_uint16 x) xa xb"
assms_2: shows "AE x in M. 0 < real_cond_exp M Invariants f x"