hcju/isabelleps huggingface dataset:
rel_compile_env_entry: fixes \<O> :: "'fun \<Rightarrow> nat option"   and \<A> :: "'m \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and f :: "'m"   and fd1 :: "('b, 'n, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'n, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef" assumes "compile_env_entry \<O> \<A> (f, fd1) = Ok (f, fd2)" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
while_inv_bot_def: fixes b :: "'c upred"   and p :: "'c upred"   and S :: "'c hrel" shows "while\<^sub>\<bottom> b invr p do S od = while\<^sub>\<bottom> b do S od"
disc_compl_moebius_ocircline: fixes M :: "moebius"   and H :: "ocircline" shows "disc_compl (moebius_ocircline M H) = moebius_pt M ` disc_compl H"
snd_msum_aform: fixes n :: "nat"   and f :: "'b \<times> 'b pdevs"   and g :: "'b \<times> 'b pdevs" shows "snd (msum_aform n f g) = msum_pdevs n (snd f) (snd g)"
assms_12: shows "path_image c1 \<inter> path_image c = {a, b}"
2: shows "YFtorNT C g maps\<^bsub>CatExp (Op C) SET\<^esub> Hom\<^bsub>C\<^esub>[\<midarrow>,dom\<^bsub>C\<^esub> g] to Hom\<^bsub>C\<^esub>[\<midarrow>,cod\<^bsub>C\<^esub> g]"
rep_eq: fixes x :: "char"   and xa :: "fin_string" shows "Rep_fin_string (fin_string_cons x xa) = string_cons x (Rep_fin_string xa)"
set_filter_single: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "{a \<in> A. a = x} = {x}"
sorted_list_of_map_inj_aux: fixes A :: "'a \<Rightarrow> 'c option"   and B :: "'a \<Rightarrow> 'c option" assumes "finite (dom A)"   and "finite (dom B)"   and "sorted_list_of_map A = sorted_list_of_map B" shows "A = B"
arg_of_real_2: fixes x :: "real" assumes "x < 0" shows "arg (complex_of_real x) = pi"
Agent_Enc: fixes X :: "agent"   and X' :: "msg"   and Y' :: "msg" shows "Messages.Agent X \<noteq> Enc X' Y'"
int_rdvd_real: fixes i :: "int"   and x :: "real" shows "(real_of_int i rdvd x) = (i dvd \<lfloor>x\<rfloor> \<and> real_of_int \<lfloor>x\<rfloor> = x)"
b2: fixes v :: "'a" shows "cblinfun_of_mat N *\<^sub>V v = basis_enum_of_vec (N *\<^sub>v vec_of_basis_enum v)"
maWW: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "(adjoint W * W) $$ (j, i) = (if j = i then 1 else 0)"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "leq1 x y"   and "leq1 y z" shows "leq1 x z"
lm033: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = injectionsUniverse \<inter> Range -` all_partitions G \<inter> Domain -` Pow N"
radical_sqrt_normal_form_sublemma: fixes a :: "real"   and b :: "real" shows "(a - b) * (a + b) = a * a - b * b"
append_imp_prefix: fixes a :: "'b list"   and b :: "'b list" shows "prefix a (a @ b)"
Theorem_15_match: fixes dsa :: "'b set"   and d :: "'b"   and ds :: "'b set" shows "Singleton_for_d.match (Singleton_for_d.gfp_F dsa d ds) = Singleton_for_d.cop dsa d ds"
all_deducible_clauses_are_finite: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" assumes "derived_clauses_are_finite R"   and "all_fulfill finite S" shows "all_fulfill finite (all_deducible_clauses R S)"
klift_def: fixes f :: "'c \<Rightarrow> 'd set" shows "bd\<^sub>\<F> f = \<mu> \<circ> \<P> f"
dsr5: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.domain_semiringl plus times one zero domain_op less_eq less" shows "domain_op (plus x y) = plus (domain_op x) (domain_op y)"
disjoint_QF: fixes k :: "nat"   and j' :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat" assumes "\<not> disjnt (QF k (j', i')) (QF k (j, i))"   and "j' \<le> k"   and "j \<le> k"   and "i' < m"   and "i < m" shows "i' = i \<and> j' = j"
absolutely_integrable_bounded_measurable_product_real: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue"   and "bounded (f ` S)"   and "g absolutely_integrable_on S" shows "(\<lambda>x. f x * g x) absolutely_integrable_on S"
bl_assm: shows "bl = bl'\<^sup>c \<and> bl' \<in># \<B>"
hom_span: fixes gens :: "(bool \<times> unit) list set" assumes "gens \<subseteq> carrier \<F>\<^bsub>{()}\<^esub>" shows "int.lift f ` \<langle>gens\<rangle>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub> = \<langle>int.lift f ` gens\<rangle>\<^bsub>int_group\<^esub>"
assms: shows "\<forall>y. P[x::=y] \<simeq> Q[x::=y]"
span_superset: fixes S :: "'c set" shows "S \<subseteq> cspan S"
assms_5: shows "deps m \<subseteq> I ` insert_messages T"
vec_hom_zero: fixes n :: "nat" shows "of_int_hom.vec_hom (0\<^sub>v n) = 0\<^sub>v n"
mat_def: shows "Matrix.mat \<equiv> map_fun id (map_fun id (map_fun id Abs_mat)) (\<lambda>nr nc f. (nr, nc, Matrix.mk_mat nr nc f))"
iim_nonempty: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and K :: "'g set" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "G \<guillemotright> K " shows "iim F G f K \<noteq> {}"
period_set_def: fixes x :: "'s" shows "period_set x = {i. 0 < i \<and> 0 < p x x i}"
min_rvar_incdec_eq_Some_min: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and x\<^sub>j :: "nat" assumes "min_rvar_incdec_eq dir s eq = Inr x\<^sub>j" shows "\<forall>x\<in>rvars_eq eq. x < x\<^sub>j \<longrightarrow> \<not> (0 < coeff (rhs eq) x \<and> \<lhd>\<^sub>u\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (UB dir s x) \<or> coeff (rhs eq) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (LB dir s x))"
V_def: shows "V \<equiv> ball (0::'a) (\<delta> / 2)"
sat_eq_reduc_sat: fixes N :: "'f set" shows "reduc_calc.saturated N = reduc_calc.reduc_saturated N"
mIH: shows "OF_same_priority_match2 \<gamma> ft p = OF_match_linear \<gamma> ft p"
unfold_rsplit_case: fixes s :: "'b prefix_match"   and u :: "'b wordinterval"   and rs :: "'b wordinterval" assumes "(Some s, u) = wordinterval_CIDR_split1 rs" shows "(case wordinterval_CIDR_split1 rs of (None, u) \<Rightarrow> [] | (Some s, u) \<Rightarrow> s # wordinterval_CIDR_split_prefixmatch u) = s # wordinterval_CIDR_split_prefixmatch u"
zero_le_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (numeral.numeral one plus n)"
recfn_wrap_Id: fixes f :: "recf"   and a :: "nat"   and n :: "nat" assumes "recfn a f" shows "recfn a (wrap_Id f n)"
measure_pmf_inject: fixes x :: "'a pmf"   and y :: "'a pmf" shows "(measure_pmf x = measure_pmf y) = (x = y)"
\<tau>reds1t_cons_\<tau>reds1t: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and es' :: "'addr expr1 list"   and xs' :: "'addr val list"   and v :: "'addr val" assumes "\<tau>reds1gt uf P t h (es, xs) (es', xs')" shows "\<tau>reds1gt uf P t h (Val v # es, xs) (Val v # es', xs')"
SUP_subset_mono: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e \<times> 'f"   and g :: "'d \<Rightarrow> 'e \<times> 'f" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod (f x) (g x)" shows "less_eq_prod (Sup_prod (f ` A)) (Sup_prod (g ` B))"
cone_Inter: fixes f :: "'c set set" assumes "\<forall>s\<in>f. cone s" shows "cone (\<Inter> f)"
poly_divides_zero: fixes p :: "'a list"   and q :: "'a list" assumes "poly p = poly []" shows "q divides p"
ap_filter_def: fixes F :: "('e \<Rightarrow> 'd) filter"   and X :: "'e filter" shows "ap_filter F X = filtermap (\<lambda>(f, x). f x) (F \<times>\<^sub>F X)"
ennreal_leI: fixes x :: "real"   and y :: "ennreal" assumes "x \<le> enn2real y" shows "ennreal x \<le> y"
f_2: shows "f \<longlonglongrightarrow> a"
inline1_in_sub_gpvs_callee: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<exists>call\<in>outs_\<I> \<I>. \<exists>s. \<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (callee s call)"
ord_option_chainD: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'c option set" assumes "Complete_Partial_Order.chain (ord_option ord) Y" shows "Complete_Partial_Order.chain ord {x. Some x \<in> Y}"
inside_out: fixes y :: "'a"   and t :: "real" assumes "y \<in> inner"   and "0 \<le> t"   and "t \<in> existence_ivl0 y"   and "flow0 y t \<in> closure outer" shows "\<exists>T>0. T \<le> t \<and> (\<forall>s\<in>{0..<T}. flow0 y s \<in> inner) \<and> flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
sumeq: shows "(\<Sum>i = 0..n. of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) = g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
cod_def: fixes f :: "('obj, 'obj) concrete_category.arr" shows "cod f = (if codomains f \<noteq> {} then SOME b. b \<in> codomains f else null)"
distrib_left_numeral: fixes v :: "num"   and b :: "'a"   and c :: "'a" shows "numeral v * (b + c) = numeral v * b + numeral v * c"
dirichlet_prod_cong: fixes f :: "nat \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and g' :: "nat \<Rightarrow> 'b" assumes "\<And>n. 0 < n \<Longrightarrow> f n = f' n"   and "\<And>n. 0 < n \<Longrightarrow> g n = g' n" shows "dirichlet_prod f g = dirichlet_prod f' g'"
singleI: fixes x :: "'a" shows "pred.eval (Predicate.single x) x"
poly_of_pm_plus: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_of_pm x (p + q) = poly_of_pm x p + poly_of_pm x q"
integral_reflect: fixes b :: "'b"   and a :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "integral (cbox (- b) (- a)) (\<lambda>x. f (- x)) = integral (cbox a b) f"
mult_less_cancel_left2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times c a) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
leg1_simps_5: shows "local.dom fg\<^sub>0h\<^sub>1.p\<^sub>1 = fg\<^sub>0h\<^sub>1.p\<^sub>1"
mem_Joints_singleton: fixes x :: "'b list"   and X :: "'b \<times> 'b pdevs" shows "(x \<in> Joints [X]) = (\<exists>y. x = [y] \<and> y \<in> Affine X)"
ide_char: fixes a :: "'a" shows "R.ide a = (R.arr a \<and> ide a)"
lookup_default_def: fixes d :: "'c"   and m :: "('d, 'c) mapping"   and k :: "'d" shows "lookup_default d m k = (case lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
const_ctxt: fixes X :: "'a state \<times> 'a state \<Rightarrow> Set(\<langle>\<langle>'b\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)" assumes "const X" shows "const (P X)"
rep_eq: fixes x :: "'a zmultiset" shows "set_antichain (frontier x) = minimal_antichain {t. 0 < zcount x t}"
left_neutral: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "mone \<odot> a = a"
not_outside_2: fixes x :: "'a"   and y :: "'a" assumes "R x y" shows "y \<in> alts'"
simps_2: fixes uw :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and ux :: "'f" shows "cond_action_ok' uw ux Yield = True"
last_action_of_thread: fixes c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and k :: "nat"   and i :: "nat" assumes "trace c 0 k" shows "(\<forall>l\<le>k. fst (c l) ! i = fst (c k) ! i) \<or> (\<exists>last<k. fst (c last) ! i \<noteq> fst (c (Suc last)) ! i \<and> (\<forall>l>last. l \<le> k \<longrightarrow> fst (c l) ! i = fst (c k) ! i))"
get_child_nodes_ok: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "known_ptr ptr"   and "type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_child_nodes ptr"
rb_aux_inv1_D1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "rb_aux_inv1 bs" shows "set bs \<subseteq> dgrad_sig_set' (length fs) dgrad"
pt_perm_supp: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
funct_def: fixes x :: "nat"   and y :: "nat" shows "funct x y = sample_uniform q \<bind> (\<lambda>s. return_spmf (s, (x * y + (q - s)) mod q))"
simps_1: fixes a :: "'a"   and b :: "'a" shows "un_MSSCons1 (mssalternate a b) = a"
ope: fixes U :: "'a set" assumes "U \<subseteq> T" shows "closedin (top_of_set S) (S \<inter> f -` U) = closedin (top_of_set T) U"
mono_neutral_left: fixes T :: "'d set"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'e)" shows "sum g S = sum g T"
Cons_3: fixes e :: "nat list \<Rightarrow> 'a" shows "lookup_base ds (tensor_vec_from_lookup ds e) is = e is"
instance_list: fixes As :: "'a list"   and SDA' :: "'a literal multiset"   and \<eta> :: "'s" assumes "negs (mset As) = SDA' \<cdot> \<eta>" shows "\<exists>As'. negs (mset As') = SDA' \<and> As' \<cdot>al \<eta> = As"
V_in_CONS: shows "learn_cons \<phi> V_constotal rmge2"
zero_eq_of_rat_iff: fixes a :: "rat" shows "((0::'d) = of_rat a) = (0 = a)"
graph_ops_5: fixes G :: "'b graph"   and E :: "('b \<times> 'b) set" shows "delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
int_and_ac_1: fixes y :: "int"   and x :: "int"   and z :: "int" shows "y AND x AND z = x AND y AND z"
find_sig_reducer_SomeD_3: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and u :: "'t"   and t :: "'a"   and i :: "nat" assumes "find_sig_reducer (map spp_of bs) u t 0 = Some i" shows "punit.lt (rep_list (bs ! i)) adds t"
ide_in_hom: fixes a :: "'a" shows "S.ide a = \<guillemotleft>a : a \<Rightarrow>\<^sub>S a\<guillemotright>"
\<Delta>'_def: shows "\<Delta>' = \<lparr>edge = edge', capacity = capacity', source = SOURCE, sink = Inner (sink \<Delta>)\<rparr>"
p2_def: shows "p2 = quadratic_part_2 var c p1"
sums_summable: fixes f :: "nat \<Rightarrow> 'c"   and l :: "'c" assumes "f sums l" shows "summable f"
ind_hom_rhom: 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 "f\<degree>\<^bsub>A,R\<^esub> \<in> rHom (A /\<^sub>r (ker\<^bsub>A,R\<^esub> f)) R"
lm06: fixes A :: "'c set"   and B :: "'d set" assumes "finite (A \<times> B) \<and> card A * card B \<le> 1" shows "trivial (A \<times> B)"
dp_total1: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_total x"
start_point_iff2: fixes x :: "'a" assumes "path x" shows "is_point (start_points x) = has_start_points x"
update_to_assert_bound_no_lhs: fixes eqlvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('i, 'a) state \<Rightarrow> ('i, 'a) state" assumes "Pivot eqlvar pivot" shows "AssertBoundNoLhs assert_bound"
max_element: fixes x :: "'a" shows "x \<le> (1::'a)\<^sup>\<omega>"
absorb_iff1: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a \<squnion> b = a)"
mat1_mult_left: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and nr :: "nat"   and nc :: "nat"   and m :: "'c list list" assumes "class.semiring_1 one times plus zero"   and "mat nr nc m" shows "mat_multI zero plus times nr (mat1I zero one nr) m = m"
fold_equality: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'a set"   and y :: "'b" assumes "comp_fun_commute f"   and "fold_graph f z A y" shows "Finite_Set.fold f z A = y"
l1_inv5_def: shows "l1_inv5 \<equiv> {s. \<forall>gny A B Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<longrightarrow> in_progressS (progress s Ra) {xnx, xgnx, xgny, xsk, xEnd} \<longrightarrow> guessed_frame Ra xgny = Some gny \<longrightarrow> can_signal s A B \<longrightarrow> 0 < signalsResp s (Running A B (Exp gny (NonceF (Ra $ nx))))}"
closure_of_subset_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "subtopology X S closure_of T \<subseteq> S"
are_operators_interfering_symmetric: fixes op\<^sub>1 :: "('a, 'b) strips_operator_scheme"   and op\<^sub>2 :: "('a, 'c) strips_operator_scheme" assumes "are_operators_interfering op\<^sub>1 op\<^sub>2" shows "are_operators_interfering op\<^sub>2 op\<^sub>1"
fun_if_subterm: fixes t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" assumes "t \<sqsubset> u" shows "\<exists>f X. u = Fun f X"
S1: shows "pr_index_enumerator n (m + 1) = comp_by_index index_of_id (pr_index_enumerator n m)"
prv_eqv_imp_transi_rev: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (eqv \<phi>1 \<phi>2)"   and "bprv (imp \<phi>1 \<chi>)" shows "bprv (imp \<phi>2 \<chi>)"
poly_eq_0_coprime: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "algebraic_semidom_class.coprime p q"   and "p \<noteq> 0"   and "q \<noteq> 0" shows "poly p x \<noteq> (0::'a) \<or> poly q x \<noteq> (0::'a)"
irsSp_Cons_ddep: fixes n'' :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var" assumes "n'' irs-ns\<rightarrow>\<^sub>d* n'"   and "n s-V\<rightarrow>\<^sub>d\<^sub>d n''"   and "n \<noteq> n''" shows "n irs-n # ns\<rightarrow>\<^sub>d* n'"
subprob_space: shows "subprob_space (distr M borel h)"
ide_def: fixes a :: "'a \<times> 'a" shows "VoV.ide a \<equiv> VoV.comp a a \<noteq> VoV.null \<and> (\<forall>f. (VoV.comp f a \<noteq> VoV.null \<longrightarrow> VoV.comp f a = f) \<and> (VoV.comp a f \<noteq> VoV.null \<longrightarrow> VoV.comp a f = f))"
fixed_points_def: shows "conjP.fixed_points = {m \<in> subgroups_of_size p. carrier (G\<lparr>carrier := P\<rparr>) \<subseteq> conjP.stabilizer m}"
l1: fixes x :: "int" assumes "\<not> QuadRes (int p) a"   and "x \<in> local.S1" shows "x \<noteq> local.f_1 x"
size_ge0: fixes kdt :: "'b kdt" shows "0 < KD_Tree.size_kdt kdt"
invertible_inverse_closed: fixes u :: "'a \<Rightarrow> 'a" assumes "invertible u"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "inverse u \<in> S \<rightarrow>\<^sub>E S"
simpler_gets_def: fixes f :: "'d \<Rightarrow> 'e" shows "gets f = (\<lambda>s. ((f s, s), False))"
zerozero_PooE: fixes x :: "nat \<times> nat" assumes "pred.eval zerozero_Poo x" obtains "zerozero x"
subs: fixes y :: "complex" assumes "y \<in> S" shows "closed_segment a y \<subseteq> S"
satPB_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "U.satPB \<xi> \<Phi> = M.satPB \<xi> \<Phi>"
eq_nat_nat_iff: fixes z :: "int"   and z' :: "int" assumes "0 \<le> z"   and "0 \<le> z'" shows "(nat z = nat z') = (z = z')"
sys_xcpts_list_def: shows "sys_xcpts_list = [NullPointer, ClassCast, OutOfMemory, ArrayIndexOutOfBounds, ArrayStore, NegativeArraySize, ArithmeticException, IllegalMonitorState, IllegalThreadState, InterruptedException]"
a: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_hoare P c c' Q = ir_valid P c c' Q"
induct: fixes P :: "'a rexp \<Rightarrow> 'a rexp \<Rightarrow> bool"   and a0 :: "'a rexp"   and a1 :: "'a rexp" assumes "\<And>r. P Zero r"   and "\<And>r s t. \<lbrakk>P r t; P s t\<rbrakk> \<Longrightarrow> P (Plus r s) t"   and "\<And>s. P One s"   and "\<And>v s. P (Atom v) s"   and "\<And>v va s. P (Times v va) s"   and "\<And>v s. P (Star v) s" shows "P a0 a1"
assms_1: shows "crdt_ops (xs @ [(oid, oper)] @ ys) deps"
principal_le_iff: fixes A :: "'b set"   and B :: "'b set" shows "(principal A \<le> principal B) = (A \<subseteq> B)"
Sort_def: fixes sort :: "'a list \<Rightarrow> 'a list" shows "Sort sort \<equiv> (\<forall>l. sorted (sort l)) \<and> (\<forall>l. sort l <~~> l)"
assms_2: shows "g \<in> quasi_measure_preserving N P"
has_codomain_iff_arr: fixes f :: "'s" shows "(codomains f \<noteq> {}) = arr f"
lc_body: fixes t :: "sterm"   and s :: "char list"   and p :: "char list" assumes "lc t"   and "s \<noteq> p" shows "body \<sigma>[s,p] t"
greatest: fixes X :: "'a set" shows "set_option (local.MaxR.MaxR_opt X) = greatest r (X \<inter> Field r)"
is_down: fixes c :: "'b word \<Rightarrow> 'c word" shows "is_down c = (LENGTH('c) \<le> LENGTH('b))"
func_def: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" shows "func f S T \<equiv> f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"
alphaInputChain: fixes xvec :: "name list"   and yvec :: "name list"   and M :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "length xvec = length yvec"   and "yvec \<sharp>* M"   and "yvec \<sharp>* P"   and "yvec \<sharp>* xvec"   and "distinct yvec" shows "inputChain xvec M P = inputChain yvec ([xvec yvec] \<bullet>\<^sub>v M) ([xvec yvec] \<bullet>\<^sub>v P)"
rcostOp_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" shows "rcostOp R I = (\<lambda>X\<in>set_rcs (b_ag R) I. \<lambda>Y\<in>set_rcs (b_ag R) I. {z. \<exists>x\<in>X. \<exists>y\<in>Y. \<exists>h\<in>I. x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<plusminus>\<^bsub>R\<^esub> h = z})"
exists_split_characterisations_1: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_2 x)"
simps_36: fixes x7 :: "'b ptrm"   and x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PSnd x7 \<noteq> PApp x31 x32"
simps: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and es :: "e list" shows "run_step d i (s, vs, es) = (let (ves, y) = split_vals_e es in case_list (s, vs, crash_error) (\<lambda>e es''. if e_is_trap e then if es'' \<noteq> [] \<or> ves \<noteq> [] then (s, vs, RSNormal [Trap]) else (s, vs, crash_error) else let (s', vs', r) = run_one_step d i (s, vs, rev ves, e) in case r of RSNormal res \<Rightarrow> (s', vs', RSNormal (res @ es'')) | _ \<Rightarrow> (s', vs', r)) y)"
bex_one_point2: fixes A :: "'c set"   and a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "(\<exists>x\<in>A. a = x \<and> P x) = (a \<in> A \<and> P a)"
eqd_pref2: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "x\<inverse>\<^sup>>u \<cdot> v = y"
a: shows "( \<Gamma> \<Rightarrow>* Em, n) \<in> derivable R*"