hcju/isabelleps huggingface dataset:
set_disconnected_nodes_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
assms: shows "\<not> liftAll (\<lambda>eX. eX \<noteq> ERR) einp"
step_set_eq: fixes ws :: "('b rexp set \<times> 'b rexp set) list"   and ps :: "('b rexp set \<times> 'b rexp set) list"   and as :: "'b list"   and ws' :: "('b rexp set \<times> 'b rexp set) list"   and ps' :: "('b rexp set \<times> 'b rexp set) list" assumes "test (ws, ps)"   and "step as (ws, ps) = (ws', ps')" shows "set ws' \<union> set ps' = set ws \<union> set ps \<union> (\<Union>a\<in>set as. {(Pderiv a (fst (hd ws)), Pderiv a (snd (hd ws)))})"
induced_function_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "A.induced_function A f = (\<lambda>x. \<Sum>i<A.return_time_function A x. f ((induced_map A ^^ i) x))"
XI: fixes xvec :: "name list"   and \<Psi>' :: "'b"   and Ra :: "('a, 'b, 'c) psi"   and Pa :: "('a, 'b, 'c) psi"   and Qa :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>'"   and "\<And>A\<^sub>R \<Psi>\<^sub>R. \<lbrakk>extractFrame Ra = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>; A\<^sub>R \<sharp>* \<Psi>'; A\<^sub>R \<sharp>* Pa; A\<^sub>R \<sharp>* Qa\<rbrakk> \<Longrightarrow> \<Psi>' \<otimes> \<Psi>\<^sub>R \<rhd> Pa \<approx> Qa" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>Pa \<parallel> Ra, \<lparr>\<nu>*xvec\<rparr>Qa \<parallel> Ra) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q)}"
Ur_def: shows "Ur = \<Union>\<^sub>\<circ> r"
SeqThrow\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>0 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>1 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
zone_set_mono: fixes A :: "('f \<Rightarrow> 'g) set"   and B :: "('f \<Rightarrow> 'g) set"   and r :: "'f list" assumes "A \<subseteq> B" shows "A\<^bsub>r \<rightarrow> 0\<^esub> \<subseteq> B\<^bsub>r \<rightarrow> 0\<^esub>"
shiftr_div_2n': fixes w :: "'a word"   and n :: "nat" shows "unat (w >> n) = unat w div 2 ^ n"
cut_greater_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>> t = {}) = (\<forall>x\<in>I. x \<le> t)"
int_ops_3: fixes n :: "num" shows "int (numeral n) = numeral n"
new2_less_sum: fixes \<phi> :: "'d ltlr" shows "size_set (new2 \<phi>) < size_set {\<phi>}"
ucast_less: fixes x :: "'b word" assumes "LENGTH('b) < LENGTH('c)" shows "ucast x < 2 ^ LENGTH('b)"
assms_2: shows "\<forall>B\<subseteq>A. inj_on Xd (f B)"
reflexive: fixes p :: "'a"   and A :: "('b, 'a) nba" shows "p \<in> reachable A p"
mk_safe_def: fixes ode_ops :: "ode_ops"   and X :: "'c set" shows "mk_safe ode_ops X = ASSERT (wd ode_ops TYPE('c)) \<bind> (\<lambda>_. safe_set ode_ops (X ::: appr_rel) \<bind> (\<lambda>s. if s then RETURN (X ::: appr_rel) else SUCCEED))"
recv_marker_changes_head_only_at_i: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration"   and i' :: "nat" assumes "distributed_system channel"   and "ev = RecvMarker i p q"   and "distributed_system.next channel trans send recv c ev c'"   and "i' \<noteq> i" shows "msgs c i' = [] \<or> hd (msgs c i') = hd (msgs c' i')"
induct: fixes x1 :: "'node"   and x2 :: "'edge list"   and x3 :: "'node"   and P :: "'node \<Rightarrow> 'edge list \<Rightarrow> 'node \<Rightarrow> bool" assumes "x1 -x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_node n \<Longrightarrow> P n [] n"   and "\<And>n as n'' as' n'. \<lbrakk>n -as\<rightarrow>\<^sub>d* n''; P n as n''; n'' -as'\<rightarrow>\<^sub>c\<^sub>d n'\<rbrakk> \<Longrightarrow> P n (as @ as') n'"   and "\<And>n as n'' V as' n'. \<lbrakk>n -as\<rightarrow>\<^sub>d* n''; P n as n''; n'' -{V}as'\<rightarrow>\<^sub>d\<^sub>d n'\<rbrakk> \<Longrightarrow> P n (as @ as') n'" shows "P x1 x2 x3"
index_image: fixes xs :: "'b list"   and X :: "'b set" assumes "distinct xs"   and "set xs = X" shows "index xs ` X = {0..<length xs}"
indmhom: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "R module N"   and "f \<in> mHom R M N" shows "f\<^sup>\<flat>\<^bsub>R M, N\<^esub> \<in> mHom R (M /\<^sub>m (ker\<^bsub>M,N\<^esub> f)) N"
Range_empty_iff: fixes r :: "('e \<times> 'd) set" shows "(Range r = {}) = (r = {})"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<sim> P \<parallel> <\<nu>x>Q"
L2P_def: fixes l :: "proj2_line" shows "L2P l \<equiv> case l of P2L p \<Rightarrow> p"
simps_4: fixes x :: "'b"   and t :: "'b tree" shows "exec (Delete x) [t] = delete x t"
tranclpD: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
measurable_outer_intervals_bounded: fixes S :: "'c set"   and a :: "'c"   and b :: "'c"   and e :: "real" assumes "S \<in> lmeasurable"   and "S \<subseteq> cbox a b"   and "0 < e" obtains \<D> :: "'c set set" where "countable \<D>" and "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" and "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>" and "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n" and "\<And>K. \<lbrakk>K \<in> \<D>; box a b \<noteq> {}\<rbrakk> \<Longrightarrow> interior K \<noteq> {}" and "S \<subseteq> \<Union> \<D>" and "\<Union> \<D> \<in> lmeasurable" and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) \<le> Sigma_Algebra.measure lebesgue S + e"
prems_2: shows "pre_subdivFace' g f r u n (ov # ovs)"
invert_side_of_triangle_1: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "C.arr h"   and "f \<cdot> g = h"   and "C.iso f" shows "C.seq (C.inv f) h \<and> g = C.inv f \<cdot> h"
singleton_contrE: fixes X :: "'a set" assumes "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> False"   and "X \<noteq> {}" obtains x :: "'a" where "X = {x}"
sorted_list_of_set_eq_Nil_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.sorted_list_of_set less_eq A = []) = (A = {})"
E: shows "sorted_list_of_map A = sorted_list_of_map B"
prems_1: shows "(a, b, c) \<in> set (x # xs)"
ChiRel_ChiStar_Source_notmem: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and T :: "'d" assumes "TS \<in> HPT ST"   and "(S, T) \<in> ChiRel (HA ST)"   and "S \<in> Conf ST"   and "T \<notin> ChiStar (HA ST) `` Source TS" shows "S \<notin> ChiStar (HA ST) `` Source TS"
h: shows "graph_homomorphism (fst R) (graph_of (X i)) f"
Lim_topological: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and net :: "'b filter" shows "(f \<longlongrightarrow> l) net = (net = bot \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in net. f x \<in> S)))"
mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "f m \<le> f n"
disc_eq_case: fixes prod :: "'a \<times> 'b" shows "(prod = prod) = (case prod of (uu_, uua_) \<Rightarrow> True)"
poly_eval_eq_poly: fixes a :: "'b \<Rightarrow> 'a"   and x :: "'b"   and p :: "'a poly" shows "poly_eval a (pm_of_poly x p) = poly p (a x)"
ideal_mult_right_in: fixes a :: "'a"   and A :: "'a set"   and b :: "'a" assumes "a \<in> More_Modules.ideal A" shows "a * b \<in> More_Modules.ideal A"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains l :: "name" and s :: "tm" and k :: "tm" and t :: "tm" and sl :: "name" and m :: "name" and n :: "name" and sm :: "name" and sn :: "name" where "atom l \<sharp> (s, k, t, sl, m, n, sm, sn)" and "atom sl \<sharp> (s, k, t, m, n, sm, sn)" and "atom m \<sharp> (s, k, t, n, sm, sn)" and "atom n \<sharp> (s, k, t, sm, sn)" and "atom sm \<sharp> (s, k, t, sn)" and "atom sn \<sharp> (s, k, t)" and "x = (s, k, t)"
assms: shows "chain R\<^sup>+\<^sup>+ xs"
simps_6: fixes c :: "'b"   and d :: "'b" shows "unify (Const c) (Const d) = (if c = d then Some [] else None)"
induct: fixes P :: "'k list \<Rightarrow> 'k list \<Rightarrow> bool"   and a0 :: "'k list"   and a1 :: "'k list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
H': shows "\<Gamma>\<turnstile>T<:(\<forall>X<:U. U')"
moebius_preserve_cross_ratio: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and M :: "moebius"   and z :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3" shows "cross_ratio (moebius_pt M z) (moebius_pt M z1) (moebius_pt M z2) (moebius_pt M z3) = cross_ratio z z1 z2 z3"
i_in_range: shows "i < length (sturm_squarefree' p) - 2"
fds_nth_zeta: shows "fds_nth fds_zeta = (\<lambda>n. if n = 0 then 0::'b else (1::'b))"
nres_inequalities_5: fixes x :: "'b" shows "SUCCEED \<noteq> RETURN x"
prems_3: shows "P,E \<turnstile> s \<surd>"
eq: shows "set (ab gs bs hs data) = set bs \<union> set hs"
f': fixes x :: "'a" assumes "x \<in> G" shows "(f has_derivative f' x) (at x within G)"
cos_double: fixes x :: "'a" shows "cos ((2::'a) * x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
Zn_iso_DirProds_prime_powers: fixes n :: "nat" assumes "n \<noteq> 0" shows "Z n \<cong> DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)"
naturality: fixes f :: "'c" assumes "C.arr f" shows "F (C.cod f) \<cdot>\<^sub>D F f = F f \<cdot>\<^sub>D F (C.dom f)"
sifum_lang_axioms: shows "sifum_lang ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars"
right_uniqueD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "right_unique A"   and "A x y"   and "A x z" shows "y = z"
assms_1: fixes q :: "'a poly" assumes "0 < degree q"   and "q \<noteq> 0"   and "q * q dvd p" shows "False"
dropWhile_append3: fixes P :: "'g \<Rightarrow> bool"   and y :: "'g"   and xs :: "'g list"   and ys :: "'g list" assumes "\<not> P y" shows "dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
arr_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.arr f = (arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f))"
sorted_ConsA: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "all_in_list (x # xs) l"   and "singleCombinators (x # xs)" shows "FWNormalisationCore.sorted (x # xs) l = (FWNormalisationCore.sorted xs l \<and> (\<forall>y\<in>set xs. smaller x y l))"
R3_H2_commute: shows "R3 \<circ> H2 = H2 \<circ> R3"
h7: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set [[1], [- 1]]"
list_insert_conv_insert: shows "equal_base.list_insert (=) = List.insert"
all1_vec_scalar_prod: fixes xs :: "real list" shows "all1_vec (length xs) \<bullet> vec_of_list xs = sum_list xs"
pe: shows "p = [:- a, 1::'a:] ^ n * q"
nprv_exuI_var: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "nprv F (subst \<phi> t x)"   and "nprv (insert (subst \<phi> (Var y) x) F) (eql (Var y) t)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "y \<notin> FvarsT t"   and "y \<notin> \<Union> (Fvars ` F)"   and "y = x \<or> y \<notin> Fvars \<phi>" shows "nprv F (exu x \<phi>)"
substT_atrm: fixes r :: "'trm"   and x :: "'var"   and t :: "'trm" assumes "r \<in> atrm"   and "x \<in> var"   and "t \<in> atrm" shows "substT r t x \<in> atrm"
Exit_not_standard_control_dependent: fixes n :: "'node" assumes "n controls\<^sub>s (_Exit_)" shows "False"
wfP_implies_wf_set_of: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfP P" shows "wf (set_of P)"
class_add_wf_mdecl: fixes wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and md :: "jvm_method mdecl"   and C :: "char list"   and cdec :: "jvm_method class" assumes "wf_mdecl wf_md P C\<^sub>0 md"   and "\<And>C\<^sub>0 md. wf_md P C\<^sub>0 md \<Longrightarrow> wf_md (class_add P (C, cdec)) C\<^sub>0 md" shows "wf_mdecl wf_md (class_add P (C, cdec)) C\<^sub>0 md"
Inf_le: fixes x :: "nat"   and A :: "nat set" assumes "x \<in> A" shows "Inf A \<le> x"
assms_2: shows "ys \<noteq> xs \<or> y \<noteq> x"
invh_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invh l"   and "invh r"   and "bheight l \<le> bheight r" shows "invh (joinL l x r)"
subset_le_dim: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> span T" shows "dim S \<le> dim T"
simps_2: fixes e :: "('b, 'c, 'd) exp"   and a :: "'d" shows "extRet2J e (RetExc a) = Throw a"
Py: shows "y \<in> F \<and> (\<forall>ya\<in>F. f y \<le> f ya)"
that: fixes f :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set \<Rightarrow> ('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "\<And>G. G \<in> Gs \<Longrightarrow> flowsto (f G - trap) T CX G"   and "Gs \<noteq> {} \<Longrightarrow> X = \<Union> (f ` Gs)" shows "thesis"
eq_neutr_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<union># b = {#}) = (a = {#} \<and> b = {#})"
attractor_set_base: fixes W :: "'a set"   and p :: "Player" shows "W \<subseteq> G'.attractor p W"
highIn_def: shows "highIn = high_in \<tau>\<^sub>1"
ispec: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and t :: "nat" assumes "\<box> t I. P t"   and "t \<in> I" shows "P t"
S4: shows "f x (b_least2 f x (Suc y)) \<noteq> 0"
mult_scalar_monomial: fixes t :: "'b"   and c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.mult_scalar (monomial c t) p = MPoly_Type_Class.punit.monom_mult c t p"
combine_eq_eval: fixes Ks :: "'a list"   and x :: "'a" shows "combine Ks (exp_base x (length Ks)) = local.eval Ks x"
tan_add: fixes x :: "'a"   and y :: "'a" assumes "cos x \<noteq> (0::'a)"   and "cos y \<noteq> (0::'a)"   and "cos (x + y) \<noteq> (0::'a)" shows "tan (x + y) = (tan x + tan y) / ((1::'a) - tan x * tan y)"
smap2_applicative: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c stream"   and y :: "'d stream" shows "smap2 f x y = Applicative.ap (Applicative.ap (pure f) x) y"
subset'_def: shows "Cardinality.subset' = (\<subseteq>)"
deriv: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_real_derivative f x) (at x)"
euler_mascheroni_bounds: fixes n :: "nat" assumes "1 \<le> n" shows "euler_mascheroni \<in> {harm n - ln (real (Suc n)) + inverse (real (2 * (n + 1)))..harm n - ln (real (Suc n)) + inverse (real (2 * n))}"
exhaust: fixes y :: "ou_witness" obtains "y = ou_witness.a"   | "y = ou_witness.b"   | "y = ou_witness.c"
isecure_total_def: shows "isecure_total \<equiv> NI_unrelated_total \<and> NI_indirect_sources_total"
finite_dimensional_vector_space_pair_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>2 :: "'f set" shows "finite_dimensional_vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2 \<equiv> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
1: shows "p = (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
hyps_3: fixes x :: "'a" assumes "x \<in> set qs" obtains t :: "'b tree" where "accs (ta_rules TA) t x"
val_Zp_ultrametric_diff: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "min (val_Zp x) (val_Zp y) \<le> val_Zp (x \<ominus> y)"
rfn_compseries_iMTr2: fixes r :: "nat"   and s :: "nat"   and xa :: "nat" assumes "0 < r"   and "0 < s"   and "xa < s * r" shows "xa div s * s \<le> r * s \<and> Suc (xa div s) * s \<le> r * s"
rep_eq: fixes x :: "'b covector" shows "Rep_covector (- x) = - Rep_covector x"
eq_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'c"   and y :: "'c" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
same_input_sig: shows "inputs (ioa.asig composition) = inputs (ioa.asig (ioa 0 id2))"
ord_gpv_the_gpvD: fixes f :: "('d, 'e, 'f) gpv"   and g :: "('d, 'e, 'f) gpv" assumes "ord_gpv f g" shows "ord_spmf (rel_generat (=) (=) (rel_fun (=) ord_gpv)) (the_gpv f) (the_gpv g)"
per_pref_ex: fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r\<^sup>\<omega>" obtains n :: "nat" where "u \<le>p r \<^sup>@ n" and "n \<noteq> 0"
finitary: shows "finite (\<Delta> ` (S \<union> N))"
supteq_def: shows "{\<unrhd>} \<equiv> {(xa, x). ??.Subterm_and_Context.supteqp xa x}"
assms_1: shows "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s cs"
Aleph_Limit: fixes \<gamma> :: "V" assumes "Limit \<gamma>" shows "\<aleph>\<gamma> = \<Squnion> (Aleph ` elts \<gamma>)"
strictify2: fixes x :: "'a"   and f :: "'a \<rightarrow> 'b" assumes "x \<noteq> \<bottom>" shows "strictify\<cdot>f\<cdot>x = f\<cdot>x"
countable: shows "countable \<^bold>E\<^bsub>bipartite_web_of \<Gamma>\<^esub>"
sf_strict_mono: fixes x :: "nat"   and y :: "nat" assumes "x < y" shows "sf x < sf y"
scan_length: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and a :: "'d" shows "length (scan f xs a) = length xs"
esl_def: shows "Opt.esl = (\<lambda>(A, r, f). (opt A, Opt.le r, Opt.sup f))"
inf_set_def: fixes A :: "'f set"   and B :: "'f set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
min_absorb2: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "order_ow U le ls"   and "y \<in> U"   and "x \<in> U"   and "le y x" shows "(with le : \<guillemotleft>min\<guillemotright> x y) = y"
one_uexpr_def: shows "1 = \<guillemotleft>1::'e\<guillemotright>"
restrict_def: fixes G :: "('d, 'e) labeled_graph" shows "restrict G = LG {(l, v1, v2). (l, v1, v2) \<in> edges G \<and> v1 \<in> vertices G \<and> v2 \<in> vertices G} (vertices G)"
T_split_min_def: fixes t :: "'b tree" shows "T_split_min t = time (split_min_tm t)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "\<phi>.A.par f f'"   and "F f = F f'" shows "f = f'"
Red_F_\<G>_def: fixes N :: "'f set" shows "wf_lift.empty_ord.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F_q q C. D \<in> Red_F_q q (wf_lift.\<G>_Fset N)}"
union_mset_add_mset_left: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "add_mset a A + B = add_mset a (A + B)"
Cons: shows "literal el literalLlist' = opposite literal el map opposite literalLlist'"
join_parity_def: fixes x :: "parity"   and y :: "parity" shows "x \<squnion> y = (if x \<sqsubseteq> y then y else if y \<sqsubseteq> x then x else Either)"
igSwapInpIPresIGWlsInp_def: fixes MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSwapInpIPresIGWlsInp MOD \<equiv> \<forall>zs z1 z2 delta inp. igWlsInp MOD delta inp \<longrightarrow> igWlsInp MOD delta (igSwapInp MOD zs z1 z2 inp)"
on_equator_in_hyp2_rep: fixes a :: "proj2" assumes "a \<in> hyp2"   and "proj2_incident a equator" shows "\<exists>x. \<bar>x\<bar> < 1 \<and> a = proj2_abs (vector [x, 0, 1])"
prime_factorization_prod_mset_primes: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b multiset" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>p. p \<in># A \<Longrightarrow> normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize (comm_monoid_mult.prod_mset times one A) = A"
vector_add_component: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(x + y) $ i = x $ i + y $ i"
alt_nondet_def: fixes m1 :: "('a, 'm) nondetT"   and m2 :: "('a, 'm) nondetT" shows "alt_nondet m1 m2 = NondetT (bind (run_nondet m1) (\<lambda>A. bind (run_nondet m2) (\<lambda>B. return (A \<^bold>\<union> B))))"
wfP_eq_minimal: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. r y z \<longrightarrow> y \<notin> Q))"
secure_def: fixes c :: "OBJ" shows "secure c = (\<forall>s ss t tt \<beta>. s \<equiv>\<^sub>\<beta> ss \<longrightarrow> s , c \<Down> t  \<longrightarrow> ss , c \<Down> tt  \<longrightarrow> (\<exists>\<gamma>. t \<equiv>\<^sub>\<gamma> tt \<and> Pbij_extends \<gamma> \<beta>))"
T1_7: fixes init :: "'c list"   and qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and x :: "'c"   and y :: "'c" assumes "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"   and "length Strat = length qs"   and "x \<noteq> y"   and "x \<in> set init"   and "y \<in> set init"   and "distinct init"   and "set qs \<subseteq> set init" shows "T\<^sub>p_opt (Lxy init {x, y}) (Lxy qs {x, y}) \<le> ALGxy_det Strat qs init x y + ALG_Pxy Strat qs init x y"