hcju/isabelleps huggingface dataset:
fall2_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and F :: "('a \<times> 'b set) stream \<Rightarrow> ('a \<times> 'b set) stream \<Rightarrow> ('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.fall2 S s0 \<delta> AP L AP' F \<equiv> Shallow.fall S s0 \<delta> AP L AP' (\<lambda>\<pi>. Shallow.fall S s0 \<delta> AP L AP' (F \<pi>))"
less_\<omega>_power: fixes \<alpha>1 :: "V"   and \<beta> :: "V"   and \<alpha>2 :: "V"   and m1 :: "nat"   and m2 :: "nat" assumes "Ord \<alpha>1"   and "Ord \<beta>"   and "\<alpha>2 \<in> elts \<alpha>1"   and "\<beta> < \<omega> \<up> \<alpha>2"   and "0 < m1"   and "0 < m2" shows "\<omega> \<up> \<alpha>2 * ord_of_nat m2 + \<beta> < \<omega> \<up> \<alpha>1 * ord_of_nat m1 + (\<omega> \<up> \<alpha>2 * ord_of_nat m2 + \<beta>)"
ext_inject: fixes CatDom :: "('a, 'c, 'e) Category_scheme"   and CatCod :: "('b, 'd, 'f) Category_scheme"   and MapM :: "'c \<Rightarrow> 'd"   and more :: "'g"   and CatDom' :: "('a, 'c, 'e) Category_scheme"   and CatCod' :: "('b, 'd, 'f) Category_scheme"   and MapM' :: "'c \<Rightarrow> 'd"   and more' :: "'g" shows "(\<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = \<lparr>CatDom = CatDom', CatCod = CatCod', MapM = MapM', \<dots> = more'\<rparr>) = (CatDom = CatDom' \<and> CatCod = CatCod' \<and> MapM = MapM' \<and> more = more')"
path_cons: fixes e :: "('d \<times> 'c) \<times> 'd \<times> 'b"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "e \<in> edges"   and "path (edge_end e) v' pth" shows "path (edge_begin e) v' (e # pth)"
assms_2: shows "f \<in> mor\<^bsub>C\<^esub>"
borel_measurable_ennreal_iff: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "((\<lambda>x. ennreal (f x)) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
type_implies_IT: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type" assumes "e \<turnstile> t : T" shows "IT t"
matrix_right_invertible_span_columns: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. A ** B = mat (1::'a)) = (vec.span (columns A) = UNIV)"
cp: shows "(\<lambda>p. Z (p ^ multiplicity p n)) = Z \<circ> (\<lambda>p. p ^ multiplicity p n)"
coinduct: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b" assumes "mono f"   and "X \<le> f (X \<squnion> gfp f)" shows "X \<le> gfp f"
vector_in_orthogonal_basis: fixes a :: "'a" assumes "a \<noteq> (0::'a)" obtains S :: "'a set" where "a \<in> S" and "(0::'a) \<notin> S" and "pairwise orthogonal S" and "independent S" and "finite S" and "span S = UNIV" and "card S = DIM('a)"
rs: shows "r = f * p\<^sup>2 + e * g * N * q\<^sup>2 \<and> s = (g - e * f) * p * q"
uint_shiftr_def: fixes x :: "uint"   and n :: "integer" shows "uint_shiftr x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined drop_bit x n else drop_bit (nat_of_integer n) x)"
set_integrable_Un: fixes M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "set_integrable M A f"   and "set_integrable M B f"   and "A \<in> sets M"   and "B \<in> sets M" shows "set_integrable M (A \<union> B) f"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "order.greater_eq a (sup b c) = (order.greater_eq a b \<and> order.greater_eq a c)"
iso_onto_image: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "Group.group H" shows "(f \<in> Group.iso G (subgroup_generated H (f ` carrier G))) = (f \<in> mon G H)"
syzygy_module_iff: fixes s :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "(s \<in> pmdl.syzygy_module B) = pmdl.represents B s 0"
p: fixes t :: "real" assumes "t \<in> {0..1}" shows "p (\<lambda>x. k (t, x))"
p: shows "map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (cmds, mem\<^sub>2) \<rightarrow>\<^bsub>sched\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
rep_eq: fixes x :: "'b blinop" shows "norm x = norm (Rep_blinop x)"
LNil_is_lappend_conv: fixes s :: "'b llist"   and t :: "'b llist" shows "(LNil = s @@ t) = (s = LNil \<and> t = LNil)"
numneg_def: fixes t :: "Cooper.num" shows "numneg t = nummul (- 1) t"
fres_def: fixes z :: "'a"   and y :: "'a" shows "z \<leftarrow> y = \<Squnion> {x. x \<cdot> y \<le> z}"
NO1: shows "OadmitFO \<sigma> ODE1 U"
assms_1: shows "run (!p) \<sigma> \<sigma>' r"
hyps: shows "alw (\<lambda>xs. \<not> P xs) \<omega>"
downset_def: shows "\<down> = \<Down> \<circ> \<eta>"
rel_carrier_Eps_in_1: fixes X :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim" shows "(SOME x. x \<in> X) \<in> X"
has_integral_cbox: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'c"   and a :: "'b"   and b :: "'b" shows "(f has_integral I) (cbox a b) = (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
min_def: fixes a :: "'c list"   and b :: "'c list" shows "prefix_order.min a b = (if prefix a b then a else b)"
per_supset_rel_def: shows "per_supset_rel \<equiv> {(p1, p2). Restr p1 (Domain p2) = p2 \<and> p1 - Domain p2 \<times> Domain p2 \<subseteq> Id}"
l6_4_2: fixes A :: "'p"   and P :: "'p"   and B :: "'p" assumes "Col A P B"   and "\<not> Bet A P B" shows "P Out A B"
lt_plus_eqI: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ord_term_strict (ordered_term.lt term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term q)" shows "ordered_term.lt term_of_pair ord_term (p + q) = ordered_term.lt term_of_pair ord_term q"
method_exit_unique: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and n :: "'h"   and n' :: "'h" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFGExit.method_exit sourcenode kind valid_edge Exit n"   and "CFGExit.method_exit sourcenode kind valid_edge Exit n'"   and "get_proc n = get_proc n'" shows "n = n'"
apex_def: shows "assoc\<mu>\<nu>\<pi>.cod.apex \<equiv> C.dom assoc\<mu>\<nu>\<pi>.cod.leg0"
ctermsDI: fixes pp :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "pp \<in> cterms \<Gamma>"   and "p \<in> dterms \<Gamma> pp" shows "p \<in> cterms \<Gamma>"
ccontr: fixes P :: "bool" assumes "\<not> P \<Longrightarrow> False" shows "P"
locally_sound\<^sub>A: shows "abs.locally_sound_mode_use \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A"
that: fixes m :: "nat" assumes "m \<le> n"   and "normalize a = p ^ m" shows "thesis"
UN_subset_iff: fixes A :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and B :: "'c set" shows "(\<Union> (A ` I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
Eq_Zero_I: fixes i :: "name"   and t :: "tm"   and H :: "fm set" assumes "insert (Var i IN t) H \<turnstile> Fls"   and "atom i \<sharp> t"   and "\<forall>B\<in>H. atom i \<sharp> B" shows "H \<turnstile> t EQ Zero"
to_mset_def: fixes t :: "('c, 'd) SkewBinomialHeap" shows "to_mset t \<equiv> BsSkewBinomialHeapStruc.bs_to_mset (Rep_SkewBinomialHeap t)"
factorization_lattice_def: fixes u :: "'b poly"   and k :: "nat"   and m :: "'b" shows "factorization_lattice u k m \<equiv> map (\<lambda>i. vec_of_poly_n (u * monom (1::'b) i) (degree u + k)) [k>..0] @ map (\<lambda>i. vec_of_poly_n (monom m i) (degree u + k)) [degree u>..0]"
jordan_block_index_2: fixes n :: "nat"   and k :: "'b" shows "dim_row (jordan_block n k) = n"
chart_last_continuous: shows "continuous_on (Collect chart_last_domainP) chart_last"
f5: shows "(- 1 * sqrt (real 2) = real_of_rat p) = (sqrt (real 2) + real_of_rat p = 0)"
reducecoeff_def: fixes t :: "MIR.num" shows "reducecoeff t = (let g = numgcd t in if g = 0 then C 0 else if g = 1 then t else reducecoeffh t g)"
mult_minus1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus one) z = uminus z"
chamber_eq_W_image: fixes C :: "'a set" assumes "chamber C" shows "\<exists>w\<in>\<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. C = w `\<rightarrow> C0"
intros_2: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and \<sigma> :: "'c \<Rightarrow> 'd" assumes "e \<sigma>" shows "ce \<sigma> (Assume e) \<sigma>"
FL_bisimilar_equivp: fixes F :: "'effect fs_set" shows "equivp (FL_bisimilar F)"
is_ref_map_def: fixes f :: "'e \<Rightarrow> 'f"   and B :: "('e, 'g) ioa"   and A :: "('f, 'g) ioa" shows "is_ref_map f B A \<equiv> (\<forall>s\<in>ioa.start B. f s \<in> ioa.start A) \<and> (\<forall>s t a. reachable B s \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> t \<longrightarrow> (\<exists>e. refines e s a t A f))"
simps_1: shows "invpst \<langle>\<rangle> = True"
Quotient_refl2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a"   and s :: "'a" assumes "Quotient R Abs Rep T"   and "R r s" shows "R s s"
method_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list" shows "method P C M \<equiv> THE (D, Ts, T, m). P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"
Var: fixes xs :: "'varSort"   and x :: "'var" shows "wls (asSort xs) (Var xs x)"
node\<^sub>i_bal_up\<^sub>i: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "split split"   and "bal_up\<^sub>i (split.node\<^sub>i k ts t)" shows "bal (Node ts t)"
nontriv_solution_imp_solution: fixes z :: "'a \<times> 'a" assumes "nontriv_solution z" shows "solution z"
step_I: fixes r :: "'a"   and r' :: "'a" assumes "I r"   and "C r"   and "B r = Some r'" shows "I r'"
less_st_def: fixes F :: "'a st"   and G :: "'a st" shows "(F < G) = (F \<le> G \<and> \<not> G \<le> F)"
in_hom_char: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (arr f \<and> ide a \<and> ide b \<and> Dom f = Dom a \<and> Cod f = Dom b)"
mkKripke_def: fixes ws :: "'f set"   and rels :: "'d \<Rightarrow> ('f \<times> 'f) set"   and val :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "mkKripke ws rels val \<equiv> \<lparr>worlds = ws, relations = \<lambda>a. Restr (rels a) ws, valuation = val\<rparr>"
homeomorphic_compactness: fixes s :: "'a set"   and t :: "'b set" assumes "s homeomorphic t" shows "compact s = compact t"
exhaust: fixes y :: "interpolation_algorithm" obtains "y = Newton"   | "y = Lagrange"   | "y = Neville_Aitken"
mult_le1: fixes y :: "V"   and x :: "V" assumes "y \<noteq> 0" shows "x \<sqsubseteq> x * y"
absolutely_integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c" assumes "continuous_on {a..b} f" shows "f absolutely_integrable_on {a..b}"
prod_n_principal_ideal: fixes R :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and J :: "nat \<Rightarrow> 'c set" assumes "Ring R"   and "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j}"   and "f \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "\<forall>k\<le>n. J k = R \<diamondsuit>\<^sub>p f k \<^bsup>\<diamondsuit>R e k\<^esup>" shows "i\<Pi>\<^bsub>R,n\<^esub> J = R \<diamondsuit>\<^sub>p mprod_expR R e f n"
obtain_range_permutation_for_two_bijections: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and f' :: "'a \<Rightarrow> 'b" assumes "bij_betw f A B"   and "bij_betw f' A B" obtains p :: "'b \<Rightarrow> 'b" where "p permutes B" and "\<forall>a\<in>A. f a = p (f' a)"
simps_3: fixes S :: "nat \<Rightarrow> typ"   and sch1 :: "type_scheme"   and sch2 :: "type_scheme" shows "$ S (sch1 =-> sch2) = $ S sch1 =-> $ S sch2"
iko: shows "is_kth_obs \<pi> (l - 1) (GREATEST i. i \<in> obs_ids \<pi> \<inter> {..<j})"
that: assumes "vsv f"   and "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "thesis"
assms_6: shows "rev.flow0 x t \<notin> K"
hom_to_Img: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "f \<in> gHom F (Img\<^bsub>F,G\<^esub> f)"
holo': fixes r :: "real"   and s :: "complex" assumes "0 \<le> r"   and "1 / 2 < Re s - r" shows "pre_newman holomorphic_on ball s r"
path_first_vertex: fixes G :: "('c, 'd) Graph_scheme"   and x :: "'c"   and xs :: "'c list" assumes "Digraph G"   and "Digraph.walk G (x # xs) \<and> distinct (x # xs)" shows "x \<notin> set xs"
content_cbox_cases: fixes a :: "'b"   and b :: "'b" shows "content (cbox a b) = (if \<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i then \<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i else 0)"
integral_std_normal_moment_abs_odd: fixes k :: "nat" shows "LBINT x. std_normal_density x * \<bar>x\<bar> ^ (2 * k + 1) = sqrt (2 / pi) * 2 ^ k * fact k"
has_derivative_transform_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and d :: "real"   and g :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g has_derivative f') (at x within s)"
space_trivial_measure: fixes X :: "'b set" shows "space (trivial_measure X) = X"
initD_def: shows "initD \<equiv> (\<lambda>_. \<infinity>)(s := 0::'c)"
subset_range_from_nat_into: fixes A :: "'a set" assumes "countable A" shows "A \<subseteq> range (from_nat_into A)"
IH: shows "(\<exists>f. t = Fun f []) \<and> (\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta>) \<or> (\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta> \<and> is_Fun m)"
A: shows "{<|v1, v2|>} = (\<lambda>(v1, v2). <|v1, v2|>) ` ({v1} \<times> {v2})"
hext_typeof_mono: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and 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 v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h'\<^esub> v = \<lfloor>T\<rfloor>"
g: shows "graph_homomorphism (snd R) (S j) g \<and> agree_on (fst R) f g"
tc_eq: fixes x :: "('a \<times> 'a) set" assumes "x ; x = x" shows "x ; x\<^sup>* = x"
that: fixes y :: "'a"   and z :: "'a" assumes "y \<in> inside (path_image p)"   and "y \<in> ball x e"   and "z \<in> outside (path_image p)"   and "z \<in> ball x e" shows "thesis"
mem_cball_0: fixes x :: "'a"   and e :: "real" shows "(x \<in> cball (0::'a) e) = (norm x \<le> e)"
decomp_H': fixes E :: "'b set"   and H :: "'b set"   and y1 :: "'b"   and y2 :: "'b"   and x' :: "'b"   and a1 :: "real"   and a2 :: "real" assumes "vectorspace E"   and "Subspace.subspace H E"   and "y1 \<in> H"   and "y2 \<in> H"   and "x' \<notin> H"   and "x' \<in> E"   and "x' \<noteq> (0::'b)"   and "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'" shows "y1 = y2 \<and> a1 = a2"
AbstrLevels_A2_A21: fixes i :: "AbstrLevelsID" assumes "sA2 \<in> AbstrLevel i" shows "sA21 \<notin> AbstrLevel i"
loopfree_digraph_axioms_def: fixes G :: "('c, 'd) pre_digraph" shows "loopfree_digraph_axioms G \<equiv> \<forall>e. e \<in> arcs G \<longrightarrow> tail G e \<noteq> head G e"
nat_filtration_comp_measurable: fixes f :: "bool stream \<Rightarrow> 'a"   and N :: "'a measure"   and n :: "nat" assumes "random_variable N f"   and "f \<circ> pseudo_proj_True n = f" shows "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"
instance_of\<^sub>l\<^sub>s_trans: fixes L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and L\<^sub>3 :: "fterm literal set" assumes "instance_of\<^sub>l\<^sub>s L\<^sub>1 L\<^sub>2"   and "instance_of\<^sub>l\<^sub>s L\<^sub>2 L\<^sub>3" shows "instance_of\<^sub>l\<^sub>s L\<^sub>1 L\<^sub>3"
cf_get_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'a"   and e :: "nat \<times> nat" shows "cf_get cf e \<equiv> assert (case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V) \<bind> (\<lambda>_. return (cf e))"
algebraic_iff_represents: fixes x :: "'b" shows "algebraic x = (\<exists>p. p represents x)"
sum_support_set: fixes pf1 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf2 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "support_set (qty_sum pf1 pf2) \<subseteq> support_set pf1 \<union> support_set pf2"
is_run_alt: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep" shows "df_system.is_run s\<^sub>0 lstep s = (s 0 = s\<^sub>0 \<and> (\<forall>i. \<exists>l. lstep l (s i) (s (Suc i))))"
disjoint_UN_iff: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "disjnt A (\<Union> (B ` I)) = (\<forall>i\<in>I. disjnt A (B i))"
bisimulation_def: fixes R :: "'s \<Rightarrow> 's \<Rightarrow> bool" shows "bisimulation R = R \<rightarrow> R"
powr_le_cancel_iff: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "1 < x" shows "(x powr a \<le> x powr b) = (a \<le> b)"
mono_bij_Inf: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono f"   and "bij f" shows "f (\<Sqinter> A) = \<Sqinter> (f ` A)"
HPhase0Read_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv5_inner s q"   and "p \<noteq> q" shows "HInv5_inner s' q"
IH_1: shows "intruder_deduct (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>) (t \<cdot> \<delta>)"
rA': shows "local.mute q k l (r, A) = (r', A')"
square_free_def: fixes p :: "'c poly" shows "square_free p = (p \<noteq> 0 \<and> (\<forall>q. 0 < degree q \<longrightarrow> \<not> q * q dvd p))"
inf_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
IH_1: fixes f :: "nat \<Rightarrow> real" assumes "\<And>x. x < count_weights True (remove_weights m1) \<Longrightarrow> f x = extract_weights True m1 x" shows "m1 = insert_weights True (remove_weights m1) f"
lt_diff: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "C \<noteq> D"
saturate_def: fixes f :: "'a \<Rightarrow> 'a" shows "saturate f = while (\<lambda>S. f S \<noteq> S) f"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_i64]"
LeastMethodDef_unfold: 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 C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" shows "(P \<turnstile> C has least M = mthd via Cs) = (MethodDefs' P C M Cs mthd \<and> LeastMethodDef_unique P C M Cs)"
lc_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" shows "lc p = lookup p (lt p)"
ex_mulDisj_prime: fixes S :: "'a set" assumes "mul_closed_set R S"   and "\<zero> \<notin> S"   and "1\<^sub>r \<in> S" shows "\<exists>mx. prime_ideal R mx \<and> S \<inter> mx = {}"
inf_par_product: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "((a::'a::refinement_lattice) \<sqinter> (b::'a::refinement_lattice)) \<parallel> ((c::'a::refinement_lattice) \<sqinter> (d::'a::refinement_lattice)) = a \<parallel> c \<sqinter> a \<parallel> d \<sqinter> b \<parallel> c \<sqinter> b \<parallel> d"
gt_iff_chop_diff_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "(t >\<^sub>t s) = (gt_chop t s \<or> gt_diff t s \<or> gt_same t s)"
semilat_def: shows "semilat = (\<lambda>(A, r, f). order r \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
slp_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.same_level_path' sourcenode targetnode kind valid_edge get_return_edges n as n' \<equiv> CFG.path sourcenode targetnode valid_edge n as n' \<and> CFG.same_level_path kind get_return_edges as"
simps_1: fixes p :: "linear_poly"   and a :: "'a" shows "ns_constraint_const (LEQ_ns p a) = a"
simps: fixes T :: "bool list set"   and r :: "bool list" shows "subtree T r = {ds \<in> T. \<exists>ds'. ds = r @ ds'}"
contk: shows "continuous_on ({0..1} \<times> {0..1}) k"
simps: fixes a :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" shows "(a \<in> directed_towards A r) = ((\<exists>x. a = x \<and> x \<in> A) \<or> (\<exists>x y. a = x \<and> y \<in> directed_towards A r \<and> (x, y) \<in> r))"
Skip: fixes G :: "prog"   and s :: "st" shows "G\<turnstile>Norm s \<midarrow>Skip\<rightarrow> Norm s"
h: shows "finprod G h (insert x F) = h x \<otimes> finprod G h F"
random_oracle_def: fixes \<sigma> :: "'a \<Rightarrow> '\<gamma> option"   and x :: "'a" shows "prf.random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> rand \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"