hcju/isabelleps huggingface dataset:
fold_filter: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'h \<Rightarrow> bool"   and xs :: "'h list" shows "fold f (filter P xs) = fold (\<lambda>x. if P x then f x else id) xs"
Rf_imp_ex_non_Rf: fixes C :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "C \<in> standard_redundancy_criterion.Rf N" shows "\<exists>CC. set_mset CC \<subseteq> N - standard_redundancy_criterion.Rf N \<and> (\<forall>I. I \<Turnstile>m CC \<longrightarrow> I \<Turnstile> C) \<and> (\<forall>C'. C' \<in># CC \<longrightarrow> C' < C)"
simps_3: fixes f :: "Core.term"   and t :: "Core.term"   and n :: "nat"   and u :: "Core.term" shows "subst_bv2 (f $ t) n u = subst_bv2 f n u $ subst_bv2 t n u"
kpower_inductl_var: fixes h :: "'c \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'd set"   and i :: "nat" assumes "h \<squnion> f \<circ>\<^sub>K g \<le> g" shows "kpower f i \<circ>\<^sub>K h \<le> g"
unique_factorization_m_factor: fixes p :: "int"   and f :: "int poly"   and g :: "int poly"   and c :: "int"   and hs :: "int poly multiset"   and n :: "nat" assumes "prime p"   and "p.unique_factorization_m (f * g) (c, hs)"   and "comm_monoid_mult_class.coprime (lead_coeff (f * g)) p"   and "poly_mod.square_free_m p (f * g)"   and "n \<noteq> 0"   and "p = p ^ n" shows "\<exists>fs gs. p.unique_factorization_m f (lead_coeff f, fs) \<and> p.unique_factorization_m g (lead_coeff g, gs) \<and> p.Mf (c, hs) = p.Mf (lead_coeff f * lead_coeff g, fs + gs) \<and> image_mset p.Mp fs = fs \<and> image_mset p.Mp gs = gs"
l1_refines_a0i_r: shows "refines (R01iar \<inter> reach a0i \<times> (l1_inv4 \<inter> l1_inv5)) med01iar a0i l1"
dom_map_option_comp: fixes g :: "'g \<Rightarrow> 'f"   and m :: "'e \<Rightarrow> 'g option" shows "dom (map_option g \<circ> m) = dom m"
is_empty_def: fixes A :: "'k multiset" shows "is_empty A = (A = {#})"
dvd_smult_cancel: fixes p :: "'c poly"   and a :: "'c"   and q :: "'c poly" assumes "p dvd smult a q"   and "a \<noteq> (0::'c)" shows "p dvd q"
in_dom_m_lookup_iff: fixes C :: "'c"   and N' :: "('c, 'd) fmap" shows "(C \<in># dom_m N') = (fmlookup N' C \<noteq> None)"
div_free_divergence_refine_4: fixes P :: "'b process" shows "(D P = {}) = DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>D P"
scalars: shows "list_all2 (\<lambda>rs ms. length rs = length ms) css hfvss"
delete_point_wf: fixes p :: "'a" shows "incidence_system (del_point p) (del_point_blocks p)"
forward_ix_forward: fixes os :: "'t list"   and s :: "'s"   and t :: "'s" shows "forward_ix os s t 0 = forward s t os"
faithful_functor_axioms: shows "faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.L"
region_def: shows "region \<equiv> map_fun Rep_scene id (\<lambda>F. {(s\<^sub>1, s\<^sub>2). \<forall>s. F s s\<^sub>1 = F s s\<^sub>2})"
is_inf_greatest: fixes x :: "'c"   and y :: "'c"   and inf :: "'c"   and z :: "'c" assumes "is_inf x y inf"   and "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> inf"
lem: fixes P :: "dB \<Rightarrow> bool"   and t :: "dB"   and n :: "nat" assumes "\<And>n ts. \<forall>t\<in>set ts. P t \<Longrightarrow> P (Var n \<degree>\<degree> ts)"   and "\<And>u ts. \<lbrakk>P u; \<forall>t\<in>set ts. P t\<rbrakk> \<Longrightarrow> P (Abs u \<degree>\<degree> ts)"   and "size t = n" shows "P t"
W6: fixes a :: "'a"   and b :: "'a" shows "-r (a l\<rightarrow> -l b) = -l (b r\<rightarrow> -r a)"
Sup_upto_llist_mono: fixes j :: "enat"   and k :: "enat"   and Xs :: "'a set llist" assumes "j \<le> k" shows "Sup_upto_llist Xs j \<subseteq> Sup_upto_llist Xs k"
mult_cancel_add_mset: fixes s :: "('b \<times> 'b) set"   and uu :: "'b"   and X :: "'b multiset"   and Y :: "'b multiset" assumes "trans s"   and "irrefl s" shows "((add_mset uu X, add_mset uu Y) \<in> mult s) = ((X, Y) \<in> mult s)"
basic: fixes A :: "'a set" assumes "A \<in> G" shows "P A"
intros_5: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
asymp_equivI': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> (1::'b)) F" shows "f \<sim>[F] g"
smooth_on_subset: fixes k :: "enat"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "k-smooth_on T f"   and "S \<subseteq> T" shows "k-smooth_on S f"
cSup_atLeastAtMost: fixes y :: "'a multiset"   and x :: "'a multiset" assumes "y \<subseteq># x" shows "Sup (subset_mset.atLeastAtMost y x) = x"
assms_2: shows "i + n < \<^bold>|w\<^bold>|"
sp_Cons: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and e :: "('c \<times> nat) edge"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 (e # es) v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> e \<in> edges g \<and> SubRel.subpath g (tgt e) es v2 subs)"
pt_fresh_bij: fixes pi :: "('e \<times> 'e) list"   and a :: "'e"   and x :: "'d" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
WHILEI_unfold: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and x :: "'c" shows "WHILE\<^bsup>I\<^esup> b f x = (if I x then if b x then f x \<bind> WHILE\<^bsup>I\<^esup> b f else RETURN x else FAIL)"
diff_add: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" shows "b - a + a = b"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs I" shows "rev_I a \<in> arcs I"
no_outstanding_write_to_read_only_memory: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "no_write_to_read_only_memory \<S> sb"
consume_top_geq: fixes ts :: "ct list"   and ts' :: "ct list"   and c_t :: "checker_type" assumes "consume (TopType ts) ts' = c_t"   and "length ts' \<le> length ts"   and "c_t \<noteq> Bot" shows "\<exists>as bs. ts = as @ bs \<and> ct_list_eq bs ts' \<and> c_t = TopType as"
Par: fixes P' :: "pi"   and Q' :: "pi"   and R' :: "pi"   and S' :: "pi" assumes "(P', Q') \<in> Rel"   and "(R', S') \<in> Rel'" shows "(P' \<parallel> R', Q' \<parallel> S') \<in> Rel''"
divides_cong_l: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x \<sim>\<^bsub>Ring_Divisibility.mult_of R\<^esub> x'"   and "x' divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> y"   and "x \<in> carrier (Ring_Divisibility.mult_of R)" shows "x divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> y"
finiteI_independent: fixes B :: "('a, 'b) vec set" assumes "vec.independent B" shows "finite B"
hom_mult: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e"   and y :: "'e" assumes "group_hom G H h"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
sigma_finite_measure_axioms: shows "sigma_finite_measure (density S Px)"
ineq2: fixes x :: "real" assumes "x \<in> {0..1}" shows "x * (1 - x)\<^sup>2 \<le> 4 / 27"
cases: fixes a :: "'form"   and i :: "'var itself" assumes "a \<in> global_assms' i" obtains v :: "'vertex" and p :: "'form" where "a = labelAtOut v (Reg p)" and "v |\<in>| vertices" and "nodeOf v = Assumption p"
axioms_2: fixes K :: "'a set"   and V :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subalgebra K V R" shows "subalgebra_axioms K V R"
evaln_elim_cases_6: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and es :: "expr list"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In3 (e # es)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains va :: "val" and s1 :: "state" and vs :: "val list" where "v = In3 (va # vs)" and "G\<turnstile>Norm s \<midarrow>e-\<succ>va\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s'"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Sup :: "'b set \<Rightarrow> 'b" assumes "class.order less_eq less"   and "class.ccpo_axioms Sup less_eq" shows "class.ccpo Sup less_eq less"
eq_refl: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x = y" shows "x \<preceq> y"
app_vrangeE: fixes b :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" obtains a :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
set_of_map_upd: fixes k :: "'a"   and v :: "'b"   and m :: "'a \<Rightarrow> 'b option" shows "set_of_map (map_upd k v m) = set_of_map (map_drop k m) \<union> {(k, v)}"
diff_add: fixes M :: "'c multiset"   and N :: "'c multiset"   and Q :: "'c multiset" shows "M - (N + Q) = M - N - Q"
super_binop_type_det: fixes op :: "super_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "super_binop_type op \<tau> \<sigma> \<rho>\<^sub>1"   and "super_binop_type op \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
prime_eq': fixes p :: "nat" shows "prime p = (1 < p \<and> (\<forall>m k. p = m * k \<longrightarrow> 1 < m \<longrightarrow> m = p))"
hd_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "xs \<noteq> []"   and "ys \<noteq> []" shows "hd (xs || ys) = (hd xs, hd ys)"
absolutely_continuousD: fixes M :: "'b measure"   and N :: "'b measure"   and A :: "'b set" assumes "absolutely_continuous M N"   and "A \<in> sets M"   and "emeasure M A = 0" shows "emeasure N A = 0"
nats_of_boolss_mod2: fixes bs :: "bool list"   and n :: "nat"   and bss :: "bool list list" assumes "length bs = n"   and "list_all (is_alph n) bss" shows "map (\<lambda>x. x mod 2) (nats_of_boolss n (bs # bss)) = map of_bool bs"
ccSUP_le_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and u :: "'c" assumes "countable A" shows "(Sup (f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
sum_list_map_num_syms_subst: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term"   and ts :: "('a, 'b) Term.term list" assumes "sum_list (map (num_syms \<circ> (\<lambda>t. t \<cdot> \<sigma>)) ts) = sum_list (map num_syms ts)" shows "\<forall>i<length ts. num_syms (ts ! i \<cdot> \<sigma>) = num_syms (ts ! i)"
h_plus_OUT_le_IN: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and i :: "nat" assumes "flow_network \<Delta> g"   and "x \<noteq> source \<Delta>" shows "d_OUT (flow_network.h_plus \<Delta> g i) x \<le> d_IN (flow_network.h_plus \<Delta> g i) x"
trisuma_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TriSumA Bet Cong A B C D E F" shows "A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F"
distinct_13: fixes x2 :: "nat"   and x41 :: "exp"   and x42 :: "exp" shows "ENat x2 \<noteq> EApp x41 x42"
real_cond_exp_int_2: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "expectation (real_cond_exp M Invariants f) = expectation f"
oprod_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "trans r"   and "trans r'"   and "antisym r"   and "antisym r'" shows "trans (r *o r')"
HomFtorContraMapsTo: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "HomC\<^bsub>C\<^esub>[f,X] maps\<^bsub>SET\<^esub> Hom\<^bsub>C\<^esub> cod\<^bsub>C\<^esub> f X to Hom\<^bsub>C\<^esub> dom\<^bsub>C\<^esub> f X"
show_matrix_real_def: fixes A :: "((real, 'c) vec, 'd) vec" shows "show_matrix_real A = (\<chi>i. show_vec_real (A $ i))"
lipschitz_on_congI: fixes L' :: "real"   and s' :: "'d set"   and g' :: "'d \<Rightarrow> 'e"   and s :: "'d set"   and L :: "real"   and g :: "'d \<Rightarrow> 'e" assumes "L'-lipschitz_on s' g'"   and "s' = s"   and "L' \<le> L"   and "\<And>x y. x \<in> s \<Longrightarrow> g' x = g x" shows "L-lipschitz_on s g"
AKcryptSK_def: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list" shows "AKcryptSK authK servK evs \<equiv> \<exists>A B tt. Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, tt\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, tt\<rbrace>\<rbrace> \<in> set evs"
cos_eq: fixes x :: "real"   and y :: "real" shows "(cos x = cos y) = (\<exists>n\<in>\<int>. x = y + 2 * n * pi \<or> x = - y + 2 * n * pi)"
poly_cmult_length: fixes a :: "'a"   and p :: "'a list" shows "length (a %* p) = length p"
transpose_def: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $h j $h i)"
Union_maximal_sets: fixes \<F> :: "'a set set" assumes "finite \<F>" shows "\<Union> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} = \<Union> \<F>"
inorder_paint: fixes c :: "color"   and t :: "('c \<times> color) tree" shows "Tree2.inorder (paint c t) = Tree2.inorder t"
TS_non_containment_causes_1: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<notin> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "set xs \<subseteq> inputs M2"   and "0 < i" shows "(\<exists>xr j. xr \<noteq> xs \<and> prefix xr xs \<and> j \<le> i \<and> vs @ xr \<in> RM M2 M1 \<Omega> V m j) \<or> (\<exists>xc. xc \<noteq> xs \<and> prefix xc xs \<and> vs @ xc \<in> C M2 M1 \<Omega> V m i - RM M2 M1 \<Omega> V m i)"
out_2: shows "gs' = filter (\<lambda>gi. \<not> p.dvdm gi (primitive_part local.g)) gs"
quasinorm_intersection_def: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 \<inter>\<^sub>N N2 = quasinorm_of (max (defect N1) (defect N2), \<lambda>f. eNorm N1 f + eNorm N2 f)"
order_iff_strict: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less" shows "less_eq a b = (less a b \<or> a = b)"
ht_at_phl_export2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "\<lbrace>x\<rbrace>y\<lbrace>\<alpha> z\<rbrace>" shows "\<lbrace>x\<rbrace>y \<cdot> \<tau> w\<lbrace>\<alpha> z \<cdot> \<tau> w\<rbrace>"
has_integral_eq_rhs: fixes f :: "'c \<Rightarrow> 'd"   and j :: "'d"   and S :: "'c set"   and i :: "'d" assumes "(f has_integral j) S"   and "i = j" shows "(f has_integral i) S"
ord_le_eq_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "prefix a b"   and "b = c" shows "prefix a c"
fact_alg_dvd_poly_higher_pderiv: fixes p :: "'a poly"   and x :: "'a"   and m :: "nat"   and k :: "nat" assumes "\<And>i. algebraic_int (poly.coeff p i)"   and "algebraic_int x"   and "m \<le> k" shows "fact m alg_dvd poly ((pderiv ^^ k) p) x"
arr_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.arr f = (arr f \<and> (ide (local.dom f) \<and> Diag (DOM (local.dom f))) \<and> ide (cod f) \<and> Diag (DOM (cod f)))"
Rf_eq_Rf_diff_Rf: fixes N :: "'a literal multiset set" shows "sr.Rf N = sr.Rf (N - sr.Rf N)"
Sup_insert: fixes a :: "'c"   and A :: "'c set" shows "Sup (insert a A) = sup a (Sup A)"
ideal_2: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "ideal I R addition multiplication zero unit"   and "a \<in> R"   and "b \<in> I" shows "multiplication b a \<in> I"
I_closed: fixes n :: "nat"   and k :: "nat" shows "liftn n \<I> k = \<I>"
ceil_divide_floor_conv: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "\<lceil>real_of_int a / real_of_int b\<rceil> = (if b dvd a then a div b else \<lfloor>real_of_int a / real_of_int b\<rfloor> + 1)"
inverse_inverse_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (inverse a) = a"
int_simps_25: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<or> #False) = P"
enc_atom_bool_take_drop_True2: fixes r :: "nat"   and I :: "('c + 'c set) list"   and p :: "'c"   and s :: "nat" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "s < length I"   and "case I ! s of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "r < s" shows "enc_atom_bool I p = take r (enc_atom_bool I p) @ True # take (s - Suc r) (drop (Suc r) (enc_atom_bool I p)) @ True # drop (Suc s) (enc_atom_bool I p)"
lub_upper: fixes Y :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain ord Y"   and "x \<in> Y" shows "ord x (lub Y)"
stc_part_Ex1: fixes x :: "complex star" assumes "x \<in> HFinite" shows "\<exists>!t. t \<in> SComplex \<and> x \<approx> t"
nat_add_left_cancel_le: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k + m \<le> k + n) = (m \<le> n)"
msg_sender_simps: fixes d :: "nat"   and sid :: "nat" shows "msg_sender (Pkt d sid) = sid"
fin_supp_1: fixes x :: "'a" shows "finite (supp x)"
chamber_pconnect: fixes x :: "'a set"   and y :: "'a set" assumes "x \<noteq> y"   and "chamber x"   and "chamber y" shows "\<exists>xs. pgallery (x # xs @ [y])"
fds_nth_zeta_pos: fixes n :: "nat" assumes "0 < n" shows "fds_nth fds_zeta n = (1::'b)"
min_le_iff_disj: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "(signed.min x y \<le>s z) = (x \<le>s z \<or> y \<le>s z)"
denote_while_n_dim: fixes \<rho> :: "complex mat"   and M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat" assumes "\<rho> \<in> carrier_mat d d"   and "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<And>\<rho>'. \<lbrakk>\<rho>' \<in> carrier_mat d d; partial_density_operator \<rho>'\<rbrakk> \<Longrightarrow> positive (DS \<rho>') \<and> trace (DS \<rho>') \<le> trace \<rho>' \<and> DS \<rho>' \<in> carrier_mat d d" shows "denote_while_n M0 M1 DS n \<rho> \<in> carrier_mat d d"
list_of_mset_empty: fixes m :: "'b multiset" shows "(list_of_mset m = []) = (m = {#})"
doubleton_eq_iff: fixes a :: "'g"   and b :: "'g"   and c :: "'g"   and d :: "'g" shows "({a, b} = {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
GuardStrip: fixes P :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f"   and g :: "'d set" assumes "P \<subseteq> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
poly_cond2E: fixes p :: "int poly" assumes "poly_cond2 p" obtains "poly_cond p" and "1 < degree p"
inv_prevs: shows "set prevs = set_prevs base (IArray.list_of tail) n"
del_list_sorted1: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted_less (xs @ a # ys)"   and "a \<le> x" shows "del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
assms: shows "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
termi_while_lemma: fixes w :: "com"   and fk :: "state"   and f :: "nat \<Rightarrow> state"   and k :: "nat"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "w \<down> fk"   and "fk = f k \<and> w = WHILE b DO c \<and> (\<forall>i. f i -c\<rightarrow> f (Suc i))" shows "\<exists>i. \<not> b (f i)"
pos_hyp: shows "(0 < poly (prod_list (retrieve_polys qs I)) x) = (z I sign = 1)"
pairs_uminus_closed: fixes x :: "'c regular"   and y :: "'c dense Filters.filter" shows "stone_phi.pairs_uminus (x, y) \<in> stone_phi.pairs"
forest_components_star: fixes x :: "'a" assumes "injective x" shows "forest_components x\<^sup>\<star> = forest_components x"
linord_helper_less_eq1_def: fixes a :: "('b, 'c) linord_helper"   and b :: "('b, 'c) linord_helper" shows "linord_helper_less_eq1 a b \<equiv> case a of LinordHelper a1 a2 \<Rightarrow> case b of LinordHelper b1 b2 \<Rightarrow> a1 < b1 \<or> a1 = b1 \<and> a2 \<le> b2"
sum_nonneg: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> (0::'b) \<le> f x" shows "(0::'b) \<le> sum f A"
ldrop_eSuc_LCons: fixes n :: "enat"   and x :: "'b"   and xs :: "'b llist" shows "ldrop (eSuc n) (x % xs) = ldrop n xs"
start_reachable: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c" shows "Graph_Start_Defs.reachable E s\<^sub>0 s\<^sub>0"
nz0: shows "x \<notin> S \<or> s x \<noteq> 0 \<or> blinfun_apply (Ds x) (f x) \<noteq> 0"
sum: fixes g :: "'d \<Rightarrow> 'b"   and S :: "'d set" shows "frechet_derivative f (at (g x)) (sum g S) = (\<Sum>a\<in>S. frechet_derivative f (at (g x)) (g a))"
in_Keys: fixes s :: "'e"   and F :: "('e \<Rightarrow>\<^sub>0 'f) set" shows "(s \<in> Keys F) = (\<exists>f\<in>F. s \<in> keys f)"
intro: fixes G :: "('c, 'd) graph_rec_scheme" assumes "fb_graph G" shows "param_DFS G"
in_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> o[F](h)) = (g \<in> o[F](h))"
vec_mat_Tensor_length: fixes xs :: "'a list"   and ys :: "'a list list" shows "length (vec_mat_Tensor xs ys) = length ys"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(qX, Y) \<in> qSwapped" shows "\<forall>qX'. Y #= qX' \<longrightarrow> qInit MOD Y = qInit MOD qX'"
simps_21: fixes x5 :: "char list" shows "Class x5 \<noteq> NT"
empty_list_set: fixes F :: "'a set" shows "{xs. set xs \<subseteq> F \<and> xs = []} = {[]}"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "well_ordered_set A less_eq" shows "well_related_set A less_eq"
lprefix_conv_lappend: fixes xs :: "'f llist"   and ys :: "'f llist" shows "lprefix xs ys = (\<exists>zs. ys = lappend xs zs)"
\<omega>_limit_set_eq: fixes x :: "'a" assumes "{0..} \<subseteq> existence_ivl0 x" shows "\<omega>_limit_set x = (\<Inter>\<tau>\<in>{0..}. closure (flow0 x ` {\<tau>..}))"
Max_wrt_among_of_weak_ranking_Cons2: fixes A :: "'alt set" assumes "x \<inter> A \<noteq> {}" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = x \<inter> A"
distinct_4: fixes x31 :: "cmd"   and x32 :: "cmd" shows "x31;; x32 \<noteq> Skip"
simps_8: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> G \<phi> = (\<forall>k. suffix k w \<Turnstile> \<phi>)"
assms_2: shows "sig_crit bs (new_syz_sigs ss bs p) p"
connected_def: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat"   and v :: "nat" shows "Graph.connected c u v \<equiv> \<exists>p. Graph.isPath c u p v"
assms_1: assumes "I = {}" shows "c \<noteq> 0"
b: shows "k \<sqsubset> length cms\<^sub>1 \<and> (cms\<^sub>1 ! k, mem\<^sub>1) \<leadsto> (cms\<^sub>1' ! k, mem\<^sub>1') \<and> cms\<^sub>1' = cms\<^sub>1[k := cms\<^sub>1' ! k]"