hcju/isabelleps huggingface dataset:
continuous_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U}))"
cp0: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2)"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "(case a1 of (a, b) \<Rightarrow> f self a b) \<tau> = (case (case a1 of (a1, a2) \<Rightarrow> \<lambda>\<tau>. (\<lambda>_. a1 \<tau>, \<lambda>_. a2 \<tau>)) \<tau> of (a, b) \<Rightarrow> f (\<lambda>_. self \<tau>) a b) \<tau>"
reflection: shows "rel_weakly_reflects_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)"
FreeGroup_uminus_closed: fixes g :: "'a freeword"   and S :: "'a set" assumes "g \<in> FreeGroup S" shows "- g \<in> FreeGroup S"
ih_21: shows "InvariantVarsM (getM state') F0 Vbl"
sin_minus_pi: fixes x :: "real" shows "sin (x - pi) = - sin x"
xor_list_left_inverse: fixes ys :: "'b list"   and xs :: "'b list" assumes "length ys \<le> length xs" shows "xs [\<oplus>] xs [\<oplus>] ys = ys"
cond_case_prod_eta: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<times> 'b \<Rightarrow> 'c" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
quad: fixes At :: "atom" shows "eval (quadratic_sub var a b c d At) (xs' @ x # xs) = aEval At ((xs' @ x # xs)[var := (A + B * sqrt C) / D])"
that_1: assumes "\<And>z. z \<in> inside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1" shows "thesis"
simps_3: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "ltl (derivation_from St) = (if is_final_dstate St then LNil else derivation_from (deterministic_RP_step St))"
intvs_ie_incdec_2: fixes l :: "int"   and h :: "int" shows "{l + 1..<h} = {l..<h} - {l}"
F_not_inj_def: shows "local.F_not_inj = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<not> inj_on f {0..<n}}"
eventually_sequentiallyI: fixes c :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "\<And>x. c \<le> x \<Longrightarrow> P x" shows "evseq P"
HoVH_char: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "HoVH \<mu>\<nu>\<tau> = B.HoHV (DN \<mu>\<nu>\<tau>)"
le_supI1: fixes x :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "x \<subseteq># a" shows "x \<subseteq># a \<union># b"
emeasure_real: fixes A :: "'a set" shows "\<exists>r\<ge>0. emeasure (lebesgue_on (cbox x y)) A = ennreal r"
main_is_initial: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" shows "initialproc fg (main fg)"
that: fixes r :: "real" assumes "q = of_real r" shows "thesis"
simps_2: fixes x :: "'d"   and xs :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c" shows "foldli (x # xs) c f \<sigma> = (if c \<sigma> then foldli xs c f (f x \<sigma>) else \<sigma>)"
assms_3: shows "g \<in> M \<rightarrow>\<^sub>M count_space A"
prime_cop_exp_poly_mod: fixes p :: "int"   and c :: "int"   and n :: "nat" assumes "prime p"   and "comm_monoid_mult_class.coprime c p"   and "n \<noteq> 0" shows "poly_mod.M (p ^ n) c \<in> {1..<p ^ n}"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated (homology_group p X) (kernel (homology_group p X) (homology_group p (discrete_topology {()})) (hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ())))) = kernel (homology_group p X) (homology_group p (discrete_topology {()})) (hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()))"
exp_cf5_def: shows "exp_cf5 \<equiv> \<lambda>x. numer_cf5 x / numer_cf5 (- x)"
p_equiv_p: fixes P :: "'b \<Rightarrow> bool" shows "(P \<longleftrightarrow> P) = true"
normalize_gives_polynomial: fixes p :: "'a list"   and K :: "'a set" assumes "set p \<subseteq> K" shows "polynomial K (R.normalize p)"
HT_simple_recursiveI: fixes R :: "('a \<times> 'a) set"   and \<pi> :: "char list \<Rightarrow> com option"   and f :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'a"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "wf R"   and "\<And>s. \<lbrakk>HT \<pi> (\<lambda>s'. (f s', f s) \<in> R \<and> P s') c Q; P s\<rbrakk> \<Longrightarrow> wp \<pi> c (Q s) s" shows "HT \<pi> P c Q"
no_sl_from_initial: fixes w :: "('b, 'c) label list"   and p :: "'b"   and c :: "'a list multiset"   and v :: "'a"   and c' :: "'a list multiset" assumes "w \<noteq> []"   and "initialproc fg p"   and "(([entry fg p], c), w, [v], c') \<in> trcl (trss fg)" shows "False"
is_final_eq: fixes s :: "nat"   and tp :: "cell list \<times> cell list" shows "is_final (s, tp) = (s = 0)"
polytope_imp_compact: fixes S :: "'a set" assumes "polytope S" shows "compact S"
ab': shows "\<langle>a, b'\<rangle> \<in>\<^sub>\<circ> s"
Polygamma_approx_ge_2_real: fixes x :: "real"   and m :: "nat"   and j :: "nat" assumes "0 < x"   and "0 < m" shows "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
empty_not_add_mset: fixes a :: "'b"   and A :: "'b multiset" shows "\<Empt> \<noteq> add_mset a A"
poincare_distance_triangle_inequality: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_distance u w \<le> poincare_distance u v + poincare_distance v w"
ccw'_sorted_scaleR: fixes xs :: "(real \<times> real) list"   and r :: "real" assumes "ccw'.sortedP 0 xs"   and "0 < r" shows "ccw'.sortedP 0 (map ((*\<^sub>R) r) xs)"
reflect_poly_pCons: fixes a :: "'a"   and p :: "'a poly" assumes "a \<noteq> (0::'a)" shows "reflect_poly (pCons a p) = Poly (rev (a # coeffs p))"
gen_lossless_gpv_continuationD: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv"   and input :: "'e" assumes "gen_lossless_gpv co \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "gen_lossless_gpv co \<I> (c input)"
bind_fv2_unchanged': fixes tm :: "Core.term"   and lev :: "nat"   and v :: "variable \<times> typ" assumes "\<not> loose_bvar tm lev"   and "bind_fv2 v lev tm = tm" shows "\<not> occs (case v of (x, xa) \<Rightarrow> Fv x xa) tm"
isNotSource_Sources: fixes i :: "AbstrLevelsID"   and S :: "CSet" assumes "isNotDSource i S" shows "\<forall>C\<in>AbstrLevel i. S \<notin> Sources i C"
vars\<^sub>s\<^sub>s\<^sub>t_unlabel_Cons: fixes l :: "'c"   and b :: "('b, 'a) stateful_strand_step"   and A :: "('c \<times> ('b, 'a) stateful_strand_step) list" shows "vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, b) # A) = vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b \<union> vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
take_bit_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (take_bit n a) = take_bit (min m n) a"
cancel: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "local.add a p\<^sub>1 p\<^sub>2 = local.add a p\<^sub>1 p\<^sub>3" shows "p\<^sub>2 = p\<^sub>3"
ord_iso_alt: fixes f :: "'c \<Rightarrow> 'd" shows "ord_iso f = (ord_embed f \<and> surj f)"
fst_combine: fixes binop :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> 'a ltln"   and \<phi> :: "'a ltln"   and i :: "enat"   and \<psi> :: "'a ltln"   and j :: "enat" shows "fst (combine binop (\<phi>, i) (\<psi>, j)) = binop (X\<^sub>n' (the_enat_0 (i - j)) \<phi>) (X\<^sub>n' (the_enat_0 (j - i)) \<psi>)"
invpst_baldR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invpst l"   and "invpst r" shows "invpst (baldR l a r)"
set_of_def: fixes m :: "'m" shows "set_of m = (\<Union>x. set (map_of m x))"
fast_bv_to_nat_Cons1: fixes bs :: "bit list"   and bin :: "num" shows "fast_bv_to_nat_helper (\<one> # bs) bin = fast_bv_to_nat_helper bs (num.Bit1 bin)"
iso_Field: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "iso r r' f" shows "f ` Field r = Field r'"
assms: fixes x :: "'a" shows "x \<in> set xs \<longrightarrow> \<not> (P x \<and> Q x)"
monom_to_monomial: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "to_fun (monom (UP Zp) a n) = monomial a n"
cs_after_all_prerecording_events: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "\<forall>i'\<ge>i. \<not> computation.prerecording_event channel trans send recv init t i'"   and "\<forall>j'<i. \<not> computation.postrecording_event channel trans send recv init t j'"   and "i \<le> length t" shows "computation.cs_equal_to_snapshot channel (distributed_system.s channel trans send recv init t i) final"
sim_r_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" shows "sim_r M M' f \<equiv> \<forall>a. \<forall>u\<in>worlds M. \<forall>v'. (f u, v') \<in> relations M' a \<longrightarrow> (\<exists>v. (u, v) \<in> relations M a \<and> f v = v')"
zcount_image_zmset: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b zmultiset"   and x :: "'a" shows "zcount (Auxiliary.image_zmset f M) x = sum (zcount M) (f -` {x} \<inter> set_zmset M)"
ltrm_smult_cring: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R"   and "p (deg R p) \<otimes> a \<noteq> \<zero>" shows "monom P ((a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p))) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = a \<odot>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)"
only_non_strict: shows "lec_rel ` cs \<subseteq> {Leq_Rel}"
chamber_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp Y (f ` C)"
gcd_diff_l: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "Arith2.gcd m n = Arith2.gcd (m - n) n"
set_disconnected_nodes_is_l_set_disconnected_nodes: shows "l_set_disconnected_nodes DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs"
Notes_Cardholder_self_False: fixes k :: "nat"   and n :: "nat"   and P :: "agent"   and C :: "agent"   and etc :: "msg"   and evs :: "event list" assumes "Notes (Cardholder k) \<lbrace>Number n, Agent P, Agent (Cardholder k), Agent C, etc\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "False"
le_less: fixes x :: "'b Word.word"   and y :: "'b Word.word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
convert_alternateNegInfinity: fixes var :: "nat"   and p :: "real mpoly"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and X :: "real \<times> real \<times> real" assumes "convert_poly var p (xs' @ x # xs) = Some X"   and "length xs' = var" shows "eval (alternateNegInfinity p var) (xs' @ x # xs) = evalUni (alternateNegInfinity' X) x"
triangle_left_implies_right: assumes "(local.inv \<eta> \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[g, f, g] \<cdot> (g \<star> local.inv \<epsilon>) = \<l>\<^sup>-\<^sup>1[g] \<cdot> \<r>[g]" shows "(f \<star> local.inv \<eta>) \<cdot> \<a>[f, g, f] \<cdot> (local.inv \<epsilon> \<star> f) = \<r>\<^sup>-\<^sup>1[f] \<cdot> \<l>[f]"
alphaAbs_qAbs_imp_alphaAbs_all_equal_or_qAFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X'"
invsL_def: fixes m :: "'h" shows "mut_m.invsL m = (\<lambda>s. mut_m.mark_object_invL m s \<and> mut_get_roots.mark_object_invL m s \<and> mut_store_ins.mark_object_invL m s \<and> mut_store_del.mark_object_invL m s \<and> mut_m.handshake_invL m s \<and> mut_m.tso_lock_invL m s)"
eSuc_plus: fixes m :: "enat"   and n :: "enat" shows "eSuc m + n = eSuc (m + n)"
h_def: shows "h = (\<lambda>(x, y). part_circlepath c r (linepath a1 a2 x) (linepath b1 b2 x) y)"
intros_4: fixes K :: "ephkey" shows "cEphK K \<in> cpayload"
sorted_wrt_singleton: fixes R :: "('b \<times> 'b) set"   and x :: "'b" shows "Linorder_Relations.sorted_wrt R [x]"
ladder_j_bound: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and index :: "nat" assumes "LeftDerivationLadder \<alpha> D L \<gamma>"   and "index < length L" shows "ladder_j L index < length (ladder_\<gamma> \<alpha> D L index)"
hom_graph_insert: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and i :: "'a" assumes "hom_graph f S x a" shows "hom_graph f (insert i S) x a"
evals_preserves_types: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E \<turnstile> es [::] Ts" shows "P,E,hp s' \<turnstile> es' [:]\<^bsub>NT\<^esub> Ts"
interaction_bounded_by_Done: fixes "consider" :: "'d \<Rightarrow> bool"   and x :: "'e"   and n :: "enat" shows "interaction_bounded_by consider (Generative_Probabilistic_Value.Done x) n"
fmpred_cases: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and x :: "'c" assumes "fmpred P m" obtains "fmlookup m x = None"   | y :: "'d" where "fmlookup m x = Some y" and "P x y"
scaleR: fixes r :: "real"   and b :: "'m" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
assms_3: fixes b :: "'a" assumes "b \<in> T" shows "\<exists>u. (\<forall>n. u n \<in> S) \<and> u \<longlonglongrightarrow> b \<and> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"
sup_left_commute: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
oreachable_init: fixes s :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> oreachable A S U"
submatrix_index: fixes i :: "nat"   and A :: "'a mat"   and I :: "nat set"   and j :: "nat"   and J :: "nat set" assumes "i < card {i. i < dim_row A \<and> i \<in> I}"   and "j < card {j. j < dim_col A \<and> j \<in> J}" shows "submatrix A I J $$ (i, j) = A $$ (pick I i, pick J j)"
sup_None2: fixes x :: "'a option" shows "x \<squnion> None = x"
Block_def: shows "Class = (\<lambda>a\<in>M. THE A. a \<in> A \<and> A \<in> Partition)"
awalk_cost_append: fixes G :: "('c, 'd) pre_digraph"   and f :: "'d \<Rightarrow> real"   and xs :: "'d list"   and ys :: "'d list" assumes "wf_digraph G" shows "wf_digraph.awalk_cost f (xs @ ys) = wf_digraph.awalk_cost f xs + wf_digraph.awalk_cost f ys"
bottomI: fixes x :: "'c" assumes "x \<sqsubseteq> \<bottom>" shows "x = \<bottom>"
infinite_super: fixes S :: "'f set"   and T :: "'f set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
periodN_D2: fixes w :: "'b list"   and n :: "nat" assumes "periodN w n" shows "n \<noteq> 0"
cases: fixes a1 :: "'Var \<Rightarrow> 'BExp set option"   and a2 :: "'AExp"   and a3 :: "'BExp set" assumes "a1 \<turnstile>\<^sub>a a2 \<in> a3" obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and e :: "'AExp" where "a1 = \<Gamma>" and "a2 = e" and "a3 = \<Union> (to_total \<Gamma> ` aexp_vars e)"
almost_full_onD: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "almost_full_on P A"   and "\<And>i. f i \<in> A" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
rel_mono_strong: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R3 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "('a, 'c, 'e) generat"   and y :: "('b, 'd, 'f) generat"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R3a :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_generat R1 R2 R3 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> generat_pures x; y1 \<in> generat_pures y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> generat_outs x; y2 \<in> generat_outs y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2"   and "\<And>z3 y3. \<lbrakk>z3 \<in> generat_conts x; y3 \<in> generat_conts y; R3 z3 y3\<rbrakk> \<Longrightarrow> R3a z3 y3" shows "rel_generat R1a R2a R3a x y"
ends_in_segment_2: fixes b :: "'b"   and a :: "'b" shows "b \<in> {a--b}"
oif_def: shows "oif = output_iface (routing_table_semantics rt (p_dst p))"
z: shows "z \<equiv> getFr [x] [] [\<phi>]"
induct: fixes x1 :: "'c llist"   and x2 :: "'c"   and P :: "'c llist \<Rightarrow> 'c \<Rightarrow> bool" assumes "Coinductive_List.lsetp x1 x2"   and "\<And>x xs. P (LCons x xs) x"   and "\<And>xs x x'. \<lbrakk>Coinductive_List.lsetp xs x; P xs x\<rbrakk> \<Longrightarrow> P (LCons x' xs) x" shows "P x1 x2"
scan_eq_nil: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" shows "(scan f xs a = []) = (xs = [])"
subgroup_is_group: assumes "Group.group H" shows "Group.group (H\<lparr>carrier := h ` carrier G\<rparr>)"
a: shows "(red F)\<^sup>*\<^sup>* (p - q) 0"
h: shows "set (get_signs (reduce_system p (qs, M_mat [[1], [- 1]] [[], [0]], [[], [0]], [[1], [- 1]]))) = set (characterize_consistent_signs_at_roots_copr p qs)"
bij_betw_number_partition_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (number_partition_of A B) ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_and_range_permutation A B) {N. number_partition (card A) N \<and> size N = card B}"
add_top: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "x + top = top"
coincide_on_lit: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and l :: "'b literal" assumes "coincide_on \<sigma> \<eta> (vars_of_lit l)" shows "subst_lit l \<sigma> = subst_lit l \<eta>"
TunerMode_CTRL_Labels_def: shows "TunerMode_CTRL_Labels = {(En ''Next'', defaultguard, defaultaction), (En ''Back'', defaultguard, defaultaction)}"
symK_bad: fixes m :: "nat"   and n :: "nat" assumes "m \<in> bad"   and "symK n = symK m" shows "n \<in> bad"
godel_finite: fixes i :: "nat"   and nl :: "nat list" shows "finite {u. Pi (Suc i) ^ u dvd godel_code' nl (Suc 0)}"
assms: shows "eWlsAbs MOD (us, s) eA"
l_zero: fixes a :: "'a" assumes "a \<in> carrier A" shows "\<zero> \<plusminus> a = a"
inj_imp_bij_betw_inv: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c set" assumes "inj f" shows "bij_betw (inv f) (f ` M) M"
assms_4: shows "\<forall>x\<in>S. a \<le> f x"
pnorm_alt_image_pset: fixes r :: "'a rexp" shows "pnorm_alt ` pset r = pset r"
spmf_eq_0_set_spmf: fixes p :: "'a spmf"   and x :: "'a" shows "(spmf p x = 0) = (x \<notin> set_spmf p)"
axiom6_F: fixes f1 :: "'h \<Rightarrow> 'l"   and f2 :: "'i \<Rightarrow> 'm"   and f3 :: "'j \<Rightarrow> 'k" shows "F2set \<circ> Fmap f1 f2 f3 = (`) f3 \<circ> F2set"
listsp_lists_eq: fixes A :: "'g set" shows "listsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> lists A)"
LIMSEQ_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
div_self: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "a div a = (1::'a)"
inverts_mat_sym: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" assumes "inverts_mat A B"   and "dim_row B = dim_col A"   and "square_mat B" shows "inverts_mat B A"
nn_integral_subalgebra2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "subalgebra M F"   and "f \<in> borel_measurable F" shows "integral\<^sup>N (restr_to_subalg M F) f = integral\<^sup>N M f"
empty_nth: fixes i :: "nat"   and x :: "'b list" assumes "\<not> i < length x" shows "x ! i = [] ! (i - length x)"
that: assumes "\<sigma>' i = \<sigma> i"   and "\<forall>j. j \<noteq> i \<longrightarrow> P (\<sigma> j) (\<sigma>' j)" shows "thesis"
subst_idemp_iff: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "(\<sigma> \<circ>\<^sub>s \<sigma> = \<sigma>) = (subst_domain \<sigma> \<inter> range_vars \<sigma> = {})"
induct: fixes x1 :: "('a, 'b, 'c) subst"   and x2 :: "('a, 'c) trm"   and P :: "('a, 'b, 'c) subst \<Rightarrow> ('a, 'c) trm \<Rightarrow> bool" assumes "TadmitF x1 x2"   and "\<And>\<sigma> \<theta>. \<lbrakk>TadmitF \<sigma> \<theta>; P \<sigma> \<theta>; TUadmit \<sigma> \<theta> UNIV\<rbrakk> \<Longrightarrow> P \<sigma> (Differential \<theta>)"   and "\<And>\<sigma> args f f'. \<lbrakk>\<And>i. TadmitF \<sigma> (args i); \<And>i. P \<sigma> (args i); SFunctions \<sigma> f = Some f'; \<And>i. dfree (Tsubst (args i) \<sigma>); TadmitFFO (\<lambda>i. Tsubst (args i) \<sigma>) f'\<rbrakk> \<Longrightarrow> P \<sigma> ($f f args)"   and "\<And>\<sigma> args f. \<lbrakk>\<And>i. TadmitF \<sigma> (args i); \<And>i. P \<sigma> (args i); SFunctions \<sigma> f = None\<rbrakk> \<Longrightarrow> P \<sigma> ($f f args)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitF \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitF \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Plus \<theta>1 \<theta>2)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitF \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitF \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Times \<theta>1 \<theta>2)"   and "\<And>\<sigma> x. P \<sigma> ($' x)"   and "\<And>\<sigma> x. P \<sigma> (Var x)"   and "\<And>\<sigma> r. P \<sigma> (Const r)" shows "P x1 x2"