hcju/isabelleps huggingface dataset:
fin_digraph_del_arc: fixes a :: "'b" shows "fin_digraph (G.del_arc a)"
a_eq_to_nat: shows "a \<noteq> mod_type_class.to_nat b"
zero_in_PPs: fixes X :: "'b set" shows "0 \<in> .[X]"
matrix_vect_scaleR: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and a :: "real"   and v :: "('b, 'd) Finite_Cartesian_Product.vec" shows "A *v a *\<^sub>R v = a *\<^sub>R (A *v v)"
b: shows "M \<longrightarrow>\<^sub>a* M'"
inverse_unique: fixes f :: "'d"   and g :: "'d" assumes "D.inverse_arrows f g" shows "D.inv f = g"
w\<^sub>h_def: shows "w\<^sub>h = fg\<^sub>0h\<^sub>1.p\<^sub>0"
uv_1: shows "x = u *\<^sub>R b + (1 - u) *\<^sub>R a"
has_integral_subset_component_le: fixes k :: "'a"   and S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and i :: "'a"   and j :: "'a" assumes "k \<in> Basis"   and "S \<subseteq> T"   and "(f has_integral i) S"   and "(f has_integral j) T"   and "\<And>x. x \<in> T \<Longrightarrow> 0 \<le> f x \<bullet> k" shows "i \<bullet> k \<le> j \<bullet> k"
tagged_partial_division_ofD_5: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "s tagged_partial_division_of i"   and "(x1, K1) \<in> s"   and "(x2, K2) \<in> s"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
vs'_def_1: shows "ves @ [a] = $$* vs' @ [v']"
assms_1: shows "is_sig_red sing_reg top_tail {f} r"
simps: fixes a :: "'b formula" shows "(a \<in> AXH) = ((\<exists>F G. a = F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F) \<or> (\<exists>F G. a = (F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> G) \<or> (\<exists>F G H. a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H) \<or> (\<exists>G H F. a = (G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H))"
exp_fg: fixes g :: "real \<Rightarrow> complex"   and g' :: "complex"   and x :: "real"   and s :: "real set"   and f :: "real \<Rightarrow> complex"   and z :: "complex" assumes "(g has_vector_derivative g') (at x within s)"   and "(f has_vector_derivative g' / (g x - z)) (at x within s)"   and "g x \<noteq> z" shows "((\<lambda>x. exp (- f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
rlex2_irrefl: fixes x :: "nat list \<times> nat list" shows "\<not> x <\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2 x"
hbex_cong: fixes A :: "hf"   and A' :: "hf"   and P :: "hf \<Rightarrow> bool"   and P' :: "hf \<Rightarrow> bool" assumes "A = A'"   and "\<And>x. x \<^bold>\<in> A' \<Longrightarrow> P x = P' x" shows "HBex A P = HBex A' P'"
prems: shows "Suc i < min (dim_row A) (dim_col A)"
nth_splice: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "n < length xs + length ys"   and "length ys \<le> length xs"   and "length xs \<le> length ys + 1" shows "splice xs ys ! n = (if even n then xs else ys) ! (n div 2)"
that: fixes i :: "nat" assumes "state_rank q\<^sub>0 n = Some i" shows "thesis"
hermitean_elems_3: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "hermitean (A, B, C, D)" shows "B = cnj C"
det_transpose: fixes A :: "('b, 'c) sq_matrix" shows "Square_Matrix.det (Square_Matrix.transpose A) = Square_Matrix.det A"
assms_3: shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (M_fin \<pi>, UNIV) w"
generate_subgroup_id: fixes H :: "'a set" assumes "subgroup H G" shows "generate G H = H"
cycle_free_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and n :: "nat"   and i :: "nat" assumes "linordered_monoid.cycle_free m n"   and "i \<le> n" shows "Le (0::'b) \<preceq> m i i"
Inf_lower2: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and u :: "'b::type"   and A :: "'b::type set"   and v :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(u::'b::type) \<in> (A::'b::type set)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (u::'b::type) (v::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (v::'b::type)"
show_defs_1: fixes d :: "nat"   and c :: "char" shows "shows_prec d c = (#) c"
hn_pass: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and x :: "'b"   and x' :: "'a" shows "hn_refine (hn_ctxt P x x') (return x') (hn_invalid P x x') P (PASS $ x)"
sup_commute: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "semilattice_sup sup less_eq less" shows "sup x y = sup y x"
is_full_functor: shows "full_functor S.comp C S.map"
inf_right_idem: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) y = inf x y"
singular_simplex_chain_map_id: fixes p :: "nat"   and X :: "'c topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" assumes "singular_simplex p X f" shows "chain_map p f (frag_of (restrict id (standard_simplex p))) = frag_of f"
fmsubset_restrict_set: fixes S :: "'c set"   and m :: "('c, 'd) fmap" shows "fmrestrict_set S m \<subseteq>\<^sub>f m"
LiftInitData_def: fixes L :: "d list" shows "LiftInitData L = Abs_data (L, DSpace)"
lnth_lappend1: fixes n :: "nat"   and xs :: "'b llist"   and ys :: "'b llist" assumes "enat n < llength xs" shows "(xs $ ys) ?! n = xs ?! n"
the_adj_chamber_facet: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "z \<lhd> the_adj_chamber C z"
TensorDiag_preserves_Diag_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u" shows "monoidal_language.Cod C (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = monoidal_language.Cod C t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> monoidal_language.Cod C u"
f: shows "f \<in> borel_measurable (stock_measure (PRODUCT t1 t2))"
manipulable_profile_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and Ri' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "social_decision_scheme agents alts sds" shows "social_decision_scheme.manipulable_profile sds R i Ri' = sds R \<prec>[SD (R i)] sds (R(i := Ri'))"
0: fixes n :: "nat" assumes "a n = \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>" shows "(\<ominus>\<^bsub>padic_int p\<^esub> a) n = \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>"
step_step': fixes S :: "(char list \<Rightarrow> 'av) option"   and C :: "(char list \<Rightarrow> 'av) option acom" shows "step (\<gamma>\<^sub>o S) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c (step' S C)"
verit_prod_simplify_2: fixes a :: "'d" shows "a * (1::'d) = a"
spmf_rel_map1: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e spmf" shows "rel_spmf R (map_spmf f x) = rel_spmf (\<lambda>x. R (f x)) x"
eventually_all_ge_at_top: fixes P :: "'b \<Rightarrow> bool" assumes "eventually P at_top" shows "\<forall>\<^sub>F x in at_top. \<forall>y\<ge>x. P y"
simps_2: fixes t :: "nat"   and volatile :: "bool"   and a :: "expr"   and e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set" shows "valid_sops_stmt t (Assign volatile a e A L R W) = (valid_sops_expr t a \<and> valid_sops_expr t e)"
assms_3: fixes u :: "'a"   and v :: "'a"   and t :: "'a" assumes "cmp u v = Lt" shows "cmp (splus t u) (splus t v) = Lt"
p_trans: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "p z y"   and "p y x" shows "p z x"
A2: shows "A2 = mat (dim_row A) nc2 (\<lambda>(i, j). four_block_mat A B C D $$ (i, j + dim_col A))"
enabled_def: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t :: "'d list" shows "enabled SES s t \<equiv> \<exists>s'. s t\<Longrightarrow>\<^bsub>SES\<^esub> s'"
measurable_restrict_space3: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> A \<rightarrow> B" shows "f \<in> restrict_space M A \<rightarrow>\<^sub>M restrict_space N B"
mset_factors_def: fixes F :: "'a multiset"   and p :: "'a" shows "mset_factors F p \<equiv> F \<noteq> {#} \<and> (\<forall>f. f \<in># F \<longrightarrow> irreducible f) \<and> p = \<Prod>\<^sub># F"
sinks_aux_subset: fixes U :: "'a set"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "U \<subseteq> sinks_aux I D U xs"
fun_belowD: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
pinf_6: fixes t :: "'b" shows "\<exists>z. \<forall>x>z. (x \<le> t) = False"
igOp_termMOD: shows "igOp termMOD = Op"
polychain_of_subsequent_eq: fixes i :: "nat"   and qs :: "'a list"   and p :: "'a" assumes "Suc i < length qs" shows "snd (polychain_of p qs ! i) = fst (polychain_of p qs ! Suc i)"
normFace_replace_in: fixes a :: "face"   and oldF :: "face"   and newFs :: "face list"   and fs :: "face list" assumes "normFace a \<in> set (normFaces (replace oldF newFs fs))" shows "normFace a \<in> set (normFaces newFs) \<or> normFace a \<in> set (normFaces fs)"
normalize_smult: fixes c :: "'a"   and p :: "'a poly" shows "normalize (smult c p) = smult (normalize c) (normalize p)"
subst_bvs1'_eq: fixes args :: "Core.term list"   and k :: "nat" assumes "args \<noteq> []" shows "subst_bvs1' (Bv k) k args = args ! 0"
dim_degree: fixes n :: "nat" shows "poly_vs.dim {x. degree x \<le> n} = n + 1"
Close1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> <\<nu>x>(P' \<parallel> Q')"
step_3: shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>R a2" obtains S :: "'procS" where "S \<in>S a1" and "\<lbrakk>S\<rbrakk> \<in>T a2"
k_invariant_frame: fixes Y :: "'mt"   and ec :: "'rep"   and obs :: "'obs"   and ec' :: "'rep"   and x :: "'rep"   and X :: "'rep list" assumes "lookup tOps Y (ec, obs) = lookup tOps Y (ec', obs)"   and "k_isNode x"   and "k_isNode ec"   and "k_isNode ec'"   and "simAbs ec' = simAbs ec" shows "lookup tOps (foldr (transUpdate x) X Y) (ec, obs) = lookup tOps (foldr (transUpdate x) X Y) (ec', obs)"
sup_inf_absorb: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "sup x (inf x y) = x"
pwd_owner_ii_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_owner_ii n)"
jkbpCSt_refl: fixes t :: "'s Trace" assumes "t \<in> Clock.jkbpC" shows "clock_sim t \<in> clock_jkbpCSt t"
removeAll_map: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and a :: "'b list" assumes "\<forall>x y. x \<noteq> y \<longrightarrow> f x \<noteq> f y" shows "removeAll (f x) (map f a) = map f (removeAll x a)"
unrest_allE: fixes P :: "'a upred" assumes "\<Sigma> \<sharp> P" obtains "P = true"   | "P = false"
simps: fixes C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list" shows "mset_ecl (C, \<sigma>) = {#mset_lit (subst_lit x \<sigma>). x \<in># mset_set (cl_ecl C)#}"
par_neq2: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B Par C D" shows "C \<noteq> D"
endofunctor_R: shows "endofunctor (\<cdot>) R"
simps_4: fixes fv :: "'a \<Rightarrow> 'b set"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and g :: "safety"   and r :: "'a Regex.regex"   and s :: "'a Regex.regex" shows "Regex.safe_regex fv safe Futu g (regex.Times r s) = ((g = Lax \<or> Regex.fv_regex fv r \<subseteq> Regex.fv_regex fv s) \<and> Regex.safe_regex fv safe Futu g s \<and> Regex.safe_regex fv safe Futu Lax r)"
normalize_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize zero = zero"
wp_fun_mono2: fixes x :: "'b"   and y :: "'b"   and p :: "'b Assertion" assumes "x \<le> y" shows "wp x p \<le> wp y p"
simps_8: shows "get_axiom Aassign = assign_axiom"
prod_mset_prime_factorization_weak: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "normalize (\<Prod>\<^sub># (prime_factorization x)) = normalize x"
inter_transfer: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (rel_set A)) (\<inter>) (\<inter>)"
length_take: fixes n :: "nat"   and xs :: "'c list" shows "length (take n xs) = min (length xs) n"
forget: fixes a :: "name"   and t1 :: "lam"   and t2 :: "lam" assumes "a \<sharp> t1" shows "t1[a::=t2] = t1"
in_face_cycle_set_other: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d"   and S :: "'d set" assumes "bidel_arc G M a"   and "S \<in> pre_digraph_map.face_cycle_sets G M"   and "{a, edge_rev M a} \<inter> S = {}" shows "S \<in> pre_digraph_map.face_cycle_sets (bidel_arc.H G M a) (bidel_arc.HM G M a)"
fun_isol: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'b" assumes "mono f"   and "g \<le> h" shows "f \<circ> g \<le> f \<circ> h"
S1_def: fixes x :: "nat"   and s1 :: "nat" shows "S1 x s1 = sample_uniform q \<bind> (\<lambda>a. sample_uniform q \<bind> (\<lambda>e1. let d1 = (x * e1 + (q - s1)) mod q in return_spmf (x, a, d1, e1)))"
exec_step_ind_Putstatic_NoField: fixes P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "\<nexists>t b. P \<turnstile> C has F,b:t in D" shows "exec_step_ind (StepI (Putstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchFieldError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
exec_step_ind_Init: fixes C :: "char list"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepC2 C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, create_init_frame P C # (stk, loc, C\<^sub>0, M\<^sub>0, pc, Called Cs) # frs, sh)"
sub_inf_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.subset_extended minus uminus inf sup bot less_eq less top" shows "inf (uminus x) (uminus y) = uminus (sup (uminus (uminus x)) (uminus (uminus y)))"
is_done: shows "\<not> snd (cs (S t i) cid) \<noteq> Done"
ge_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.non_strict_order less_eq"   and "less_eq y x"   and "less_eq z y" shows "less_eq z x"
intros_1: fixes x :: "comb" shows "x \<Rrightarrow>\<^sup>1 x"
is_set_category: shows "set_category SC.comp (cardSuc (cmax |UNIV| natLeq))"
diag_charpoly: shows "charpoly A *\<^sub>S 1 = X ^ (n + 1) *\<^sub>S \<^bold>C (B n) + (\<Sum>i<n. X ^ (i + 1) *\<^sub>S \<^bold>C (B i - A * B (i + 1))) - \<^bold>C (A * B 0)"
internal_case_prod_conv: fixes c :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
proj_psi_def: shows "proj_psi = outer_prod \<psi> \<psi>"
is_pos_neg_not_is_pos: fixes L :: "'b literal" shows "is_pos (- L) = is_neg L"
bspec: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
chain_lprefix_lSup: fixes A :: "'c llist set"   and xs :: "'c llist" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "xs \<in> A" shows "xs \<sqsubseteq> lSup A"
remdups_ex2: fixes xs :: "'a list" assumes "1 < length (remdups xs)" shows "\<exists>a\<in>set xs. \<exists>b\<in>set xs. a \<noteq> b"
f3: shows "\<exists>y\<in>carrier rec_rng_of_frac. y \<otimes>\<^bsub>rec_rng_of_frac\<^esub> x = \<one>\<^bsub>rec_rng_of_frac\<^esub> \<and> x \<otimes>\<^bsub>rec_rng_of_frac\<^esub> y = \<one>\<^bsub>rec_rng_of_frac\<^esub>"
assms_3: shows "is_partial_heap cmp heap l (m + 1) r"
poly_x_minus_y_def: fixes p :: "'b poly" shows "poly_x_minus_y p = poly_lift p \<circ>\<^sub>p x_y"
path_linepath: fixes a :: "'a"   and b :: "'a" shows "path (linepath a b)"
covector_comp_inf: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "top \<sqinter> z = z" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
assms_2: fixes x :: "'a" shows "inres S x = inres S' x"
span_eq_iff: fixes s :: "'b set" shows "(span s = s) = subspace s"
unit_in_hom_2: shows "\<guillemotleft>\<phi> : src\<^sub>D (F f) \<Rightarrow>\<^sub>D g' \<star>\<^sub>D F f\<guillemotright>"
borel_measurable_nn_integral: fixes f :: "'a \<Rightarrow> (nat \<Rightarrow> val) \<Rightarrow> ennreal"   and N :: "'a measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M Pi\<^sub>M (set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs)) (\<lambda>x. stock_measure (\<Gamma> x)))" shows "(\<lambda>xa. integral\<^sup>N (Pi\<^sub>M (set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs)) (\<lambda>x. stock_measure (\<Gamma> x))) (f xa)) \<in> borel_measurable N"
complement_character: fixes x :: "i" assumes "CMC"   and "\<exists>w. \<not> O w x" shows "\<forall>w. P w (\<midarrow> x) = (\<not> O w x)"
dvd_power: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "0 < n \<or> x = one" shows "dvd.dvd times x (power.power one times x n)"
divisor_count_bound_gen: fixes f :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. 2 \<le> f n" shows "\<forall>\<^sub>F n in sequentially. real (divisor_count n) < 2 powr ((ln (real n) + 8 / ln 2 * f n * ln (ln (real n))) / ln (f n))"
count: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "count s' \<le> eSuc (count s)"
Im_complex_div_gt_0: fixes a :: "complex"   and b :: "complex" shows "(0 < Im (a / b)) = (0 < Im (a * cnj b))"
no_msg_change_if_no_event: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isSend ev \<longrightarrow> getId ev \<noteq> i"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "msgs c i = msgs c' i"
sinh_real_at_top: shows "filterlim sinh at_top at_top"
A0: shows "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
sq_mtx_transpose_transpose: fixes A :: "'a sq_mtx" shows "A\<^sup>\<dagger>\<^sup>\<dagger> = A"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (src a)"
le_INF_iff: fixes u :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(u \<le> \<Sqinter> (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
disjoint_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set set" assumes "inj_on f (\<Union> A)"   and "disjoint A" shows "disjoint ((`) f ` A)"
strict_mono_sequence_partition: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "strict_mono f"   and "f 0 \<le> x"   and "filterlim f at_top sequentially" shows "\<exists>k. x \<in> {f k..<f (Suc k)}"
part_def: fixes v :: "'a"   and \<R> :: "'a set set" shows "[v]\<^sub>\<R> \<equiv> THE R. R \<in> \<R> \<and> v \<in> R"
deg_shifts_pp_def: fixes xs :: "'x list"   and d :: "nat"   and fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "deg_shifts_pp xs d fs = concat (map (\<lambda>f. map (\<lambda>t. monom_mult_punit (1::'b) t f) (deg_le_sect_pp xs (d - poly_deg_pp f))) fs)"
assms_5: shows "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root"
prems_4: shows "\<C> = (s_inst \<S> ! j)\<lparr>local := local (s_inst \<S> ! j) @ tvs, label := arb_labs, return := arb_return\<rparr>"
CARAT_DERIV: fixes f :: "'b \<Rightarrow> 'b"   and l :: "'b"   and x :: "'b" shows "(f has_field_derivative l) (at x) = (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
set_cases_2: fixes e :: "'b"   and a :: "('c, 'b) arr" assumes "e \<in> set2_arr a" obtains z1 :: "'c" and z2 :: "'c" where "a = MkArr z1 z2 e"
lr_fps_denominator'_def: fixes cs :: "'a list" shows "lr_fps_denominator' cs = Poly cs"
real_of_float_of_int_eq: fixes z :: "int" shows "real_of_float (float_of_int z) = real_of_int z"
restrict_return_spmf: fixes x :: "'c"   and A :: "'c set" shows "return_spmf x \<upharpoonleft> A = (if x \<in> A then return_spmf x else return_pmf None)"
split: fixes P :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and prod :: "'f \<times> 'g" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
dg_prod_Dom_app: fixes x :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "dg_prod I \<AA>\<lparr>Dom\<rparr>\<lparr>x\<rparr> = (\<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Dom\<rparr>\<lparr>x\<lparr>i\<rparr>\<rparr>)"
bij_betw_exp_times_self_atLeastAtMost: fixes a :: "real"   and b :: "real" assumes "- 1 \<le> a"   and "a \<le> b" shows "bij_betw (\<lambda>x. x * exp x) {a..b} {a * exp a..b * exp b}"
all_chains_def: fixes p :: "'c \<Rightarrow> bool"   and \<Gamma> :: "'d \<Rightarrow> 'c option"   and rs :: "'c" shows "all_chains p \<Gamma> rs \<equiv> p rs \<and> (\<forall>l rs. \<Gamma> l = Some rs \<longrightarrow> p rs)"