hcju/isabelleps huggingface dataset:
backtrack_simplex'_def: shows "backtrack_simplex' = backtrack_cs backtrack_nsc_code"
C_def: fixes x :: "real"   and y :: "real" shows "C x y = blinfun_of_matrix ((\<lambda>_ _. 0)((1, 0) := (\<lambda>_. 0)((1, 0) := C11 x y, (0, 1) := C12 x y), (0, 1) := (\<lambda>_. 0)((1, 0) := C21 x y, (0, 1) := C22 x y)))"
\<pi>_def: shows "\<pi> = (\<lambda>i. if i < n then m + i else if i < m + n then i - n else i)"
fup2: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b" shows "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x"
AE_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "f \<in> borel_measurable M" shows "almost_everywhere (density M f) P = (AE x in M. 0 < f x \<longrightarrow> P x)"
cd_path_shift: fixes j :: "nat"   and k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "j \<le> k"   and "is_path \<pi>" shows "(i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (i - j cd\<^bsup>\<pi> \<guillemotleft> j\<^esup>\<rightarrow> k - j)"
all_edge_cases: shows "\<gamma> = (\<lambda>x. twoC (0, x)) \<or> \<gamma> = (\<lambda>y. twoC (y, 1)) \<or> \<gamma> = (\<lambda>x. twoC (1, x)) \<or> \<gamma> = (\<lambda>y. twoC (y, 0))"
valid: shows "valid (fst (iterate t step ([(f, Partial_Recursive.list_decode x, [])], None)))"
borel_eq_atMost: shows "borel = sigma UNIV (range atMost)"
FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: fixes i :: "nat" shows "FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = (\<lambda>s. Mprefix (Tr\<^sub>f i s) (\<lambda>e. FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i (Up\<^sub>f i s e)))"
that: assumes "R x y"   and "P x"   and "Q y" shows "thesis"
add_neg_numeral_special_4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) (numeral.numeral one plus n) = neg_numeral.sub minus plus one n num.One"
NTUadmit_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) trm"   and \<theta> :: "('b + 'a, 'c) trm"   and U :: "('c + 'c) set" shows "NTUadmit \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
\<G>\<F>_elim: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> \<G>\<F> \<phi> w" shows "w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"
map_option_is_None: fixes f :: "'e \<Rightarrow> 'd"   and opt :: "'e option" shows "(map_option f opt = None) = (opt = None)"
if_True: fixes x :: "'b"   and y :: "'b" shows "(if True then x else y) = x"
snd_def: fixes prod :: "'b \<times> 'a" shows "snd prod = (case prod of (x1, x2) \<Rightarrow> x2)"
comm_ring_hom_axioms: shows "comm_ring_hom poly_lift"
Pair_mono: fixes x :: "'a"   and x' :: "'a"   and y :: "'b"   and y' :: "'b" assumes "x \<le> x'"   and "y \<le> y'" shows "(x, y) \<le> (x', y')"
Rep_seqauto_tuple: fixes SA :: "('a, 'b, 'c) seqauto" shows "Rep_seqauto SA = (States SA, InitState SA, Labels SA, Delta SA)"
CO_trans: fixes cl :: "'b potype" assumes "CL cl" shows "trans (order cl)"
mult_set_def: fixes A :: "rat set"   and B :: "rat set" shows "mult_set A B = {w. \<exists>x\<in>A. \<exists>y\<in>B. w = x * y}"
sub_def: fixes P :: "pol"   and Q :: "pol" shows "P \<langle>-\<rangle> Q = P \<langle>+\<rangle> neg Q"
left_minus: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) a = zero"
rev_predicate2D: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'f"   and y :: "'g"   and Q :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "P x y"   and "P \<le> Q" shows "Q x y"
Red1NewArray: fixes i :: "32 word"   and h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and T :: "ty"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "0 \<le>s i"   and "(h', a) \<in> allocate h (Array_type T (nat (sint i)))" shows "uf,P,t \<turnstile>1 \<langle>newA T\<lfloor>Val (Intg i)\<rceil>,(h, l)\<rangle> -\<lbrace>NewHeapElem a (Array_type T (nat (sint i)))\<rbrace>\<rightarrow> \<langle>addr a,(h', l)\<rangle>"
less_linear: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b \<or> a = b \<or> b \<prec>\<^bsub>D\<^esub> a"
monoI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
vsubstAbs_preserves_good: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "goodAbs A" shows "goodAbs (A $[x1 // x]_xs)"
vcg_disj_lift: fixes P :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and P' :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and Q' :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"   and "\<lbrace>P'\<rbrace> c \<lbrace>Q'\<rbrace>" shows "\<lbrace>\<lambda>s. P s \<or> P' s\<rbrace> c \<lbrace>\<lambda>s. Q s \<or> Q' s\<rbrace>"
vderiv: shows " D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0<--<t - r / 2}"
interaction_bounded_by_map_gpv': fixes "consider" :: "'d \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'd"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and f :: "'b \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'c" assumes "interaction_bounded_by (consider \<circ> g) gpv n" shows "interaction_bounded_by consider (map_gpv' f g h gpv) n"
ltrm_of_sum_diff_degree: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q < deg R p" shows "monom (UP R) ((p \<oplus>\<^bsub>UP R\<^esub> q) (deg R (p \<oplus>\<^bsub>UP R\<^esub> q))) (deg R (p \<oplus>\<^bsub>UP R\<^esub> q)) = monom (UP R) (p (deg R p)) (deg R p)"
f_join_empty: fixes xs :: "'b list" shows "xs \<Join>\<^sub> {} = []"
circ_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<circ> \<le> y\<^sup>\<circ>"
power_impr: fixes b :: "'a"   and a :: "'a"   and n :: "nat" assumes "b r\<rightarrow> a = a" shows "b ^ n r\<rightarrow> a = a"
MP_thin: fixes HA :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and HB :: "fmla set"   and H :: "fmla set" assumes "HA \<turnstile> A IMP B"   and "HB \<turnstile> A"   and "HA \<union> HB \<subseteq> H" shows "H \<turnstile> B"
le_shiftr1': fixes u :: "'b word"   and v :: "'b word" assumes "shiftr1 u \<le> shiftr1 v"   and "shiftr1 u \<noteq> shiftr1 v" shows "u \<le> v"
ideal_generated_def: fixes S :: "'b set" shows "ideal_generated S = \<Inter> {I. ideal I \<and> S \<subseteq> I}"
right_unique_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "right_unique R = (R\<inverse>\<inverse> OO R \<le> (=))"
onll_step_invariantE: fixes A :: "('e \<times> ('f, 'g, 'h, 'i) seqp, 'j) automaton"   and I :: "'j \<Rightarrow> bool"   and \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "('e \<times> 'i) \<times> 'j \<times> 'e \<times> 'i \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('f, 'g, 'h, 'i) seqp"   and a :: "'j"   and \<xi>' :: "'e"   and p' :: "('f, 'g, 'h, 'i) seqp"   and l :: "'i"   and l' :: "'i" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "I a"   and "l \<in> labels \<Gamma> p"   and "l' \<in> labels \<Gamma> p'" shows "P ((\<xi>, l), a, \<xi>', l')"
bit_numeral_Bit1_Suc_iff: fixes m :: "num"   and n :: "nat" shows "bit (numeral (num.Bit1 m)) (Suc n) = bit (numeral m) n"
simps: fixes a :: "('form, 'var) out_port"   and n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" shows "(a \<in> hyps_for' n p) = (\<exists>h. a = h \<and> hyps n h = Some p)"
eval_elim_cases_5: fixes G :: "prog"   and s :: "st"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In3 []\<succ>\<rightarrow> (v, s')" obtains "v = In3 []" and "s' = Norm s"
len_empty_on_subview2: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "\<parallel>len v ts c\<parallel> = 0 \<and> v=v1\<parallel>v2 \<longrightarrow> \<parallel>len v2 ts c\<parallel> = 0"
eq_iff: fixes x :: "'b"   and y :: "'b" shows "([:x:] = [:y:]) = (x = y)"
pow_combine_def: fixes A :: "'c set"   and B :: "'c set set" shows "pow_combine A B \<equiv> (\<Union>X\<in>Pow A. \<Union>Y\<in>B. {X \<union> Y}) \<union> Pow A"
hadamard_on_i_def: fixes i :: "nat" shows "hadamard_on_i i = partial_state2.pmat_extension dims {i} (vars1 - {i}) hadamard"
row_length_vect_mat: fixes v :: "'a list"   and m :: "'a list list" shows "row_length (vec_mat_Tensor v m) = length v * row_length m"
comp_arr_dom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and a :: "'c" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a" shows "C f a = f"
if_polynomial_0_rank: fixes w :: "nat \<Rightarrow> real" assumes "polynomial_f w \<noteq> 0" shows "r ^ N_half \<le> cprank (A w)"
While: fixes P :: "'d set"   and I :: "'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 b :: "'d set"   and c :: "('d, 'e, 'f) com"   and V :: "('d \<times> 'd) set"   and A :: "'d set"   and Q :: "'d set" assumes "P \<subseteq> I"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> I \<inter> b) c ({t. (t, \<sigma>) \<in> V} \<inter> I),A"   and "I \<inter> - b \<subseteq> Q"   and "wf V" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P whileAnno b I V c Q,A"
finite_power_lookup_not_one: fixes f :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" shows "finite {x. f x ^ lookup t x \<noteq> (1::'c)}"
dom_eqI': fixes a :: "'c"   and f :: "'c" assumes "a \<in> Cop.domains f" shows "a = Cop.dom f"
le_zero_eq: fixes n :: "'q multiset" shows "(n \<subseteq># {#}) = (n = {#})"
simps_3: fixes uu :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and v :: "'a" shows "option_assn uu (Some v) None = false"
select_convs_5: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "eps \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = eps"
utp_theory_continuous_axioms: shows "utp_theory_continuous \<H>"
Par_Wbis_discrL2: fixes indis :: "('a \<times> 'a) set"   and c2 :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com" assumes "PL_Indis indis"   and "(c2, d) \<in> PL_Indis.Wbis tval aval indis"   and "PL_Indis.discr tval aval indis c1" shows "(c1 | c2, d) \<in> PL_Indis.Wbis tval aval indis"
assms_1: fixes S :: "'a set" assumes "S \<in> \<F>" shows "open S"
squarefree_prod_coprime: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> coprime (f a) (f b)"   and "\<And>a. a \<in> A \<Longrightarrow> squarefree (f a)" shows "squarefree (prod f A)"
octo_of_real_of_int_eq: fixes z :: "int" shows "octo_of_real (real_of_int z) = octo_of_int z"
comp_def: fixes g :: "'d"   and f :: "'d" shows "Fop.Aop.comp g f \<equiv> Dop.comp f g"
integral_real_density: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (f x))) g = LINT x|M. f x * g x"
axioms_1: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "consequence_relation Bot entails"
gchoose_rising_sum: fixes n :: "int"   and k :: "nat" assumes "0 \<le> n" shows "(\<Sum>j = 0..n. j + int k gchoose k) = n + int k + 1 gchoose Suc k"
123: fixes x :: "'a" shows "- - x \<sqinter> x = x"
induct: fixes P :: "nat \<Rightarrow> ('a \<times> nat) list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "('a \<times> nat) list" assumes "\<And>s. P s []"   and "\<And>s f d xs. \<lbrakk>d \<le> s \<Longrightarrow> P (s - d) xs; \<not> d \<le> s \<Longrightarrow> P d xs\<rbrakk> \<Longrightarrow> P s ((f, d) # xs)" shows "P a0 a1"
exhaust: fixes y :: "'b acom" obtains x1 :: "'b" where "y = SKIP {x1}"   | x21 :: "char list" and x22 :: "aexp" and x23 :: "'b" where "y = x21 ::= x22 {x23}"   | x31 :: "'b acom" and x32 :: "'b acom" where "y = x31;; x32"   | x41 :: "bexp" and x42 :: "'b acom" and x43 :: "'b acom" and x44 :: "'b" where "y = IF x41 THEN x42 ELSE x43 {x44}"   | x51 :: "'b" and x52 :: "bexp" and x53 :: "'b acom" and x54 :: "'b" where "y = {x51} WHILE x52 DO x53 {x54}"
plus_in_float_intervalI: fixes a :: "real"   and A :: "float interval"   and b :: "real"   and B :: "float interval" assumes "a \<in>\<^sub>r A"   and "b \<in>\<^sub>r B" shows "a + b \<in>\<^sub>r A + B"
prems_3: shows "\<not> ps (S t i) p \<noteq> None"
alternating_list_append_1: fixes m :: "nat"   and s :: "'b"   and t :: "'b"   and n :: "nat" assumes "gcd_nat.greater_eq m 2" shows "alternating_list m s t @ alternating_list n s t = alternating_list (m + n) s t"
coprime_normalize_right_iff: 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"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "algebraic_semidom.coprime times one a (normalize b) = algebraic_semidom.coprime times one a b"
star_rtc1: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x\<^sup>\<star> + x \<le> x\<^sup>\<star>"
comp_associative: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times (times x y) z = times x (times y z)"
ide_Ide: fixes a :: "'c \<times> 'c" assumes "C.ide (fst a) \<and> C.ide (snd a)" shows "C.VxV.ide a"
rem_implicit_pres_is_standard_operator': fixes n :: "char list"   and p :: "(nat \<times> nat) list"   and es :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and c :: "nat" assumes "is_standard_operator (n, p, es, c)" shows "is_standard_operator' (rem_implicit_pres (n, p, es, c))"
inv_equality: fixes y :: "complex vec"   and x :: "complex vec" assumes "y + x = 0\<^sub>v d"   and "x \<in> carrier_vec d"   and "y \<in> carrier_vec d" shows "\<ominus>\<^bsub>V\<^esub> x = y"
map_div_is_smult_inverse: fixes a :: "'c"   and p :: "'c poly" shows "map_poly (\<lambda>x. x / a) p = Polynomial.smult (inverse a) p"
node_net_state': fixes s :: "(nat \<Rightarrow> 'd) \<times> 'e net_state"   and i :: "nat"   and T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "s \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U" shows "\<exists>\<sigma> \<zeta> R. s = (\<sigma>, NodeS i \<zeta> R)"
insort_nth2: fixes xs :: "'b list"   and a :: "'b"   and i :: "nat" assumes "sorted xs"   and "a \<notin> set xs"   and "i < length xs"   and "index (insort a xs) a \<le> i"   and "xs \<noteq> []" shows "insort a xs ! Suc i = xs ! i"
simps: fixes P :: "'a prog"   and a1 :: "char list"   and a2 :: "((char list \<times> char list) \<times> staticb \<times> ty) list" shows "(P \<turnstile> a1 has_fields a2) = ((\<exists>C D fs ms FDTs FDTs'. a1 = C \<and> a2 = FDTs' \<and> class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> C \<noteq> Object \<and> P \<turnstile> D has_fields FDTs \<and> FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs) \<or> (\<exists>D fs ms FDTs. a1 = Object \<and> a2 = FDTs \<and> class P Object = \<lfloor>(D, fs, ms)\<rfloor> \<and> FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs))"
Field_ofilter: fixes r :: "('d \<times> 'd) set" assumes "wo_rel r" shows "ofilter r (Field r)"
padic_set_def: fixes p :: "int" shows "padic_set p = {f. (\<forall>m. f m \<in> carrier (residue_ring (p ^ m))) \<and> (\<forall>n m. m < n \<longrightarrow> residue (p ^ m) (f n) = f m)}"
chain_map_sum: fixes I :: "'c set"   and p :: "nat"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'e) \<Rightarrow>\<^sub>0 int" assumes "finite I" shows "chain_map p g (sum f I) = sum (chain_map p g \<circ> f) I"
initial: fixes a :: "'d set"   and i :: "'c set"   and t :: "'d \<Rightarrow> 'c \<Rightarrow> 'c set"   and c :: "'c \<Rightarrow> bool" shows "initial (nba a i t c) = i"
some_geodesic_segment_description_1: fixes x :: "'b"   and y :: "'b"   and S :: "'b set" assumes "\<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S" shows "geodesic_segment_between {x--S--y} x y"
isl_map_sum: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and s :: "'g + 'h" shows "isl (map_sum f g s) = isl s"
Fract_cases: fixes q :: "'b fract" obtains a :: "'b" and b :: "'b" where "q = Fract a b" and "b \<noteq> (0::'b)"
cyclic_group_axioms_def: fixes G :: "('b, 'c) cyclic_group_scheme" shows "cyclic_group_axioms G \<equiv> \<^bold>g\<^bsub>G\<^esub> \<in> carrier G \<and> carrier G \<subseteq> range (([^]\<^bsub>G\<^esub>) \<^bold>g\<^bsub>G\<^esub>)"
prv_imp_refl: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla" shows "prv (imp \<phi> \<phi>)"
splits_wrt_image_sum_list_fst_subset: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "splits_wrt (ps, qs) T F" shows "sum_list ` listset (map cone ps) \<subseteq> ideal F \<inter> P[X]"
rel_compp: fixes R1 :: "'d \<Rightarrow> 'j \<Rightarrow> bool"   and S1 :: "'j \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'k \<Rightarrow> 'h \<Rightarrow> bool"   and R3 :: "'f \<Rightarrow> 'l \<Rightarrow> bool"   and S3 :: "'l \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_generat (R1 OO S1) (R2 OO S2) (R3 OO S3) = rel_generat R1 R2 R3 OO rel_generat S1 S2 S3"
ccSup_union_distrib: fixes A :: "'a set"   and B :: "'a set" assumes "countable A"   and "countable B" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
Heap_eqI: fixes f :: "'a Heap"   and g :: "'a Heap" assumes "\<And>h. execute f h = execute g h" shows "f = g"
hyps_1: shows "(red (G - {p}))\<^sup>*\<^sup>* p y"
llist_of_stream_unfold_stream: fixes SHD :: "'d \<Rightarrow> 'c"   and STL :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "llist_of_stream (unfold_stream SHD STL x) = unfold_llist (\<lambda>_. False) SHD STL x"
fw_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "i \<le> n"   and "j \<le> n"   and "i' \<le> n"   and "j' \<le> n" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i j i' j') (m i' j')"
cosh_dist_def: fixes a :: "proj2"   and b :: "proj2" shows "cosh_dist a b \<equiv> (sqrt (exp_2dist a b) + sqrt (1 / exp_2dist a b)) / 2"
unit_factor_ratfps_def: fixes x :: "'b ratfps" shows "unit_factor x = ratfps_shift (ratfps_subdegree x) x"
listDiffDoubleRemoveAll: fixes x :: "'b"   and a :: "'b list"   and b :: "'b list" assumes "x \<in> set a" shows "list_diff b a = list_diff b (x # a)"
assms: fixes i :: "'a"   and j :: "'b" shows "(F *\<^sub>V ket i) \<bullet>\<^sub>C ket j = ket i \<bullet>\<^sub>C (G *\<^sub>V ket j)"
verts_union: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" shows "verts (Digraph_Component.union G H) = verts G \<union> verts H"
f_simps_1: fixes n :: "nat"   and x :: "'v"   and y :: "'v" shows "f n (\<langle>x\<rangle>, \<langle>y\<rangle>) = (if edge \<Gamma> x y then h 0 (x, y) - h n (x, y) else if edge \<Gamma> y x then h n (y, x) - h 0 (y, x) else 0)"
wf_action_order: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "wfP (action_order E)\<^sup>\<noteq>\<^sup>\<noteq>"
rel_G_mono': fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and x :: "('n, 'p, 'r, 't, 'v, 'x, 'z) G"   and y :: "('o, 'q, 's, 'u, 'w, 'y, 'z) G"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool" assumes "rel_G L1 L2 Co1 Co2 Contra1 Contra2 x y"   and "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_G L1' L2' Co1' Co2' Contra1' Contra2' x y"
mult_poly_raw_def: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" shows "PAC_Polynomials.mult_poly_raw p q = (\<Sum>y\<in>#p. mult_poly_by_monom y q)"
assms_2: shows "\<forall>A\<in>F. A \<subseteq> E \<and> s \<le> card A \<and> card A \<le> k"