hcju/isabelleps huggingface dataset:
hyps: fixes u :: "symbol list"   and v :: "symbol list"   and w :: "symbol list" assumes "LeftDerivation (u @ v) D w"   and "derivation_ge D (length u)" shows "LeftDerivation v (derivation_shift D (length u) 0) (drop (length u) w)"
assms_2: shows "eval\<^sub>c\<^sub>s F G Cs"
natural_transformation_\<T>e: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) (Y a') (\<T> e)"
cs_not_nil: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" shows "cs\<^bsup>\<pi>\<^esup> n \<noteq> []"
OPT2_B2: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) = real (length qs - 1) / 2"
set_bits_bit_eq: fixes a :: "'a" shows "set_bits (bit a) = a"
assms_4: shows "lead_coeff b \<in> Units (R\<lparr>carrier := K\<rparr>)"
in_hom_def: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright> \<equiv> a \<in> Cop_S.domains f \<and> b \<in> Cop_S.codomains f"
lunit_char_1: fixes f :: "'a" assumes "ide f" shows "arr (PB.lunit f) \<and> src f \<in> sources (PB.lunit f) \<and> trg f \<in> targets (PB.lunit f)"
Gcd_singleton: fixes a :: "'a" shows "Gcd {a} = normalize a"
arcsin_le_arcsin: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x \<le> y"   and "y \<le> 1" shows "arcsin x \<le> arcsin y"
flip_append1: fixes n :: "nat"   and xs :: "bool list"   and ys :: "bool list" assumes "n < length xs" shows "flip (xs @ ys) n = flip xs n @ ys"
arccos_unique: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> pi"   and "cos x = y" shows "arccos y = x"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "times_hom hom" shows "semigroup_mult_hom hom"
a_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
app_iso_simps_1: fixes hom :: "('e, 'f, 'c, 'd) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" shows "verts (app_iso hom G) = iso_verts hom ` verts G"
depth_fn: fixes A :: "'b trm"   and x :: "'b"   and T :: "PreSimplyTyped.type" shows "depth A < depth (Fn x T A)"
group_part: fixes f :: "'d \<Rightarrow> 'e"   and merge :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d list \<Rightarrow> 'f"   and p :: "'d"   and xs :: "'d list" assumes "groupsort f merge g" shows "groupsort.group_part f merge p xs = (rev (filter (\<lambda>x. f x < f p) xs), fold merge (filter (\<lambda>x. f x = f p) xs) p, rev (filter (\<lambda>x. f p < f x) xs))"
less_imp_le: fixes x :: "'b list"   and y :: "'b list" assumes "x <p y" shows "x \<le>p y"
dom_char: fixes f :: "'a" shows "local.dom f = C.cod f"
exhaust_sel: fixes stream :: "'b stream" obtains "stream = shd stream ## stl stream"
infinite_coinduct: fixes X :: "'b set \<Rightarrow> bool"   and A :: "'b set" assumes "X A"   and "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})" shows "infinite A"
simps: fixes l :: "Vars.var list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "add_dummies_conf l (\<Gamma>, e, S) = (\<Gamma>, e, S @ map Dummy (rev l))"
simps_1: fixes A :: "'a" shows "atom (Pos A) = A"
power_0: fixes a :: "'e" shows "a ^ 0 = (1::'e)"
toStates_inv_2: fixes steps :: "step list"   and ss :: "(integer \<times> edge list) iarray"   and start :: "edgeIndex"   and lbls :: "(String.literal, nat) assoc_list" assumes "toStates steps = (ss, start, lbls)" shows "0 < IArray.length ss"
fmrel_cases: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and x :: "'e" assumes "fmrel P m n" obtains "fmlookup m x = None" and "fmlookup n x = None"   | a :: "'c" and b :: "'d" where "fmlookup m x = Some a" and "fmlookup n x = Some b" and "P a b"
simps_18: fixes V :: "'b"   and a :: "'c"   and e :: "('a, 'b, 'c) exp" shows "fv (insync\<^bsub>V\<^esub> (a) e) = fv e"
eucl_rel_poly_iff: fixes x :: "'a poly"   and y :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "eucl_rel_poly x y (q, r) = (x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y))"
natural_transformation_axioms: shows "natural_transformation C D G Adj.GFG.map (\<eta> \<circ> G)"
trunc_bl2bin: fixes m :: "nat"   and bl :: "bool list" shows "take_bit m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
CCFix_nonrec_eqvt: fixes \<pi> :: "perm"   and cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var"   and e :: "exp" shows "\<pi> \<bullet> CoCallArityAnalysis.CCfix_nonrec cccExp x e = CoCallArityAnalysis.CCfix_nonrec (\<pi> \<bullet> cccExp) (\<pi> \<bullet> x) (\<pi> \<bullet> e)"
\<omega>_sum_less_\<omega>_power: fixes \<alpha> :: "V"   and \<alpha>s :: "V list" assumes "sorted_wrt (\<lambda>x y. y \<le> x) (\<alpha> # \<alpha>s)"   and "list.set (\<alpha> # \<alpha>s) \<subseteq> ON" shows "\<omega>_sum \<alpha>s < \<omega> \<up> \<alpha> * \<omega>"
prv_exi_cong: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "bprv (imp \<phi> \<chi>)" shows "bprv (imp (exi x \<phi>) (exi x \<chi>))"
mult_aformE: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "(real \<times> real pdevs) \<times> real"   and y :: "real"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e X"   and "y \<in> aform_err e Y" shows "x * y \<in> aform_err e (mult_aform X Y)"
acc_le_prodI: fixes r\<^sub>A :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "acc r\<^sub>A"   and "acc r\<^sub>B" shows "acc (Product.le r\<^sub>A r\<^sub>B)"
set_pmf_def: shows "set_pmf \<equiv> map_fun measure_pmf id (\<lambda>M. {x. Sigma_Algebra.measure M {x} \<noteq> 0})"
len_lower_bound: fixes n :: "nat" assumes "0 < n" shows "2 ^ (length (nat_to_bv n) - Suc 0) \<le> n"
sorted_list_of_set_3: 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" shows "distinct (linorder.sorted_list_of_set less_eq A)"
gr_zeroI: fixes n :: "'e multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
o_def: fixes f :: "'n \<Rightarrow> 'm"   and g :: "'l \<Rightarrow> 'n" shows "f \<circ> g = (\<lambda>x. f (g x))"
simps_17: shows "get_axiom ADIGeq = DIGeqaxiom"
Sup_upper: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
pnz: shows "p \<noteq> 0\<^sub>p"
index_map_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and f :: "'b \<Rightarrow> 'c" assumes "i < dim_row A"   and "j < dim_col A" shows "map_mat f A $$ (i, j) = f (A $$ (i, j))"
mult_nonneg_nonpos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times a b) zero"
dvd: shows "H $$ (0, 0) dvd H $$ (i, 0)"
WT_resource_attach: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and res :: "('c, 'd) resource" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>' \<turnstile>res res \<surd>" shows "\<I> \<turnstile>res conv \<rhd> res \<surd>"
walk_decomp_2: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (xs @ ys)" shows "Graph.walk G ys"
getParts_def: fixes rs :: "'a simple_rule list" shows "getParts rs = partitioningIps (extract_IPSets rs) [wordinterval_UNIV]"
get_host_locs_def: shows "get_host_locs = \<Union> (range get_shadow_root_locs) \<union> (\<Union>ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
path_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "path (reversepath g) = path g"
Re_sum: fixes f :: "'a \<Rightarrow> quat"   and S :: "'a set" shows "quat.Re (sum f S) = (\<Sum>x\<in>S. quat.Re (f x))"
fair_def: fixes s :: "'s"   and t :: "'s" shows "fair s t = (\<lambda>xs. alw (ev (HLD {s})) xs \<longrightarrow> alw (ev (\<lambda>xs. HLD {s} xs \<and> nxt (HLD {t}) xs)) xs)"
catalan_aux_correct: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "catalan_aux n k (catalan k) = catalan n"
simps_2: fixes seq :: "('d, 'e, 'f) com \<Rightarrow> ('d, 'e, 'f) com \<Rightarrow> ('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list" shows "sequence seq (c # cs) = (case cs of [] \<Rightarrow> c | a # list \<Rightarrow> seq c (sequence seq cs))"
oInv_bound2: fixes a :: "ordinal" shows "a < F (oSuc (oInv F a))"
expands_to_cong: fixes f :: "real \<Rightarrow> real"   and F :: "'a"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "(f expands_to F) basis"   and "\<forall>\<^sub>F x in at_top. f x = g x" shows "(g expands_to F) basis"
fact'_3: fixes x :: "'b" assumes "p \<noteq> 0"   and "ipoly p x = (0::'b)" shows "\<exists>!q. q \<in> set qs \<and> ipoly q x = (0::'b)"
A: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
apply_outputs_literal: fixes P :: "'a aexp list"   and r :: "nat"   and v :: "value"   and s :: "'a \<Rightarrow> value option" assumes "P ! r = L v"   and "r < length P" shows "apply_outputs P s ! r = Some v"
Var_preserves_good: fixes xs :: "'varSort"   and x :: "'var" shows "good (Var xs x)"
insert: fixes a :: "'a"   and F :: "'a multiset" assumes "a \<in># A"   and "P F" shows "P (add_mset a F)"
preserves_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>D (F \<mu>) = map\<^sub>0 (src\<^sub>C \<mu>)"
grounding_of_cls_def: fixes C :: "'a literal multiset" shows "\<G>_F C = {C \<cdot> \<sigma> |\<sigma>. is_ground_subst \<sigma>}"
pos_1: fixes i :: "nat" shows "0 \<le> f i"
poly_of_int_poly: fixes p :: "int poly"   and x :: "int" shows "poly (of_int_poly p) (of_int x) = of_int (poly p x)"
sets_eq: shows "sets (dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. M \<sigma> \<bind> (\<lambda>y. return (state_measure (shift_var_set (V \<union> V')) (case_nat t \<Gamma>)) (case_nat y \<sigma>)))) = sets (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) (case_nat undefined \<rho>))"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "ManiPref'.dX X d = {x \<in> X. Xd x = d}"
strict_order_iff: fixes a :: "'d"   and b :: "'d" shows "(a < b) = (a = inf a b \<and> a \<noteq> b)"
Par2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto> a<\<nu>x> \<prec> (P \<parallel> Q')"
flow_not: fixes s :: "real" assumes "0 < s"   and "s \<le> t" shows "flow0 x s \<notin> P"
cases: fixes a :: "sterm" assumes "is_value a" obtains cs :: "(Term_Class.term \<times> sterm) list" where "a = Sabs cs"   | vs :: "sterm list" and name :: "name" where "a = name $$ vs" and "list_all is_value vs" and "name |\<in>| C"
elims_2: fixes x :: "'a"   and xa :: "'a \<Rightarrow> 'b"   and xb :: "('a \<Rightarrow> 'b option) \<times> ('a \<Rightarrow> 'b option)" assumes "in_bounds x xa xb" obtains xc :: "'a" and v :: "'a \<Rightarrow> 'b" and lb :: "'a \<Rightarrow> 'b option" and ub :: "'a \<Rightarrow> 'b option" where "x = xc" and "xa = v" and "xb = (lb, ub)" and "v xc \<ge>\<^sub>l\<^sub>b lb xc \<and> v xc \<le>\<^sub>u\<^sub>b ub xc"
step_\<G>: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl"   and \<nu> :: "'b set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P step \<phi> \<nu>"
ii: shows "\<nexists>L. L \<in> C \<and> complement L \<in> PP2"
a_adopt_node_locs_def: fixes remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr" shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs parent owner_document document_ptr = (if parent = None then {} else remove_child_locs (the parent) owner_document) \<union> set_disconnected_nodes_locs document_ptr \<union> set_disconnected_nodes_locs owner_document"
hyps: fixes cs :: "'edge list" shows "upd_cs cs (as' @ [a]) = a # upd_cs cs as'"
i'_range: shows "ix p d < (i + 1) * 2 ^ (lv p d - l) \<and> (i - 1) * 2 ^ (lv p d - l) < ix p d"
Solution_imp_subdprodl: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Solutions" shows "subdprodl x y"
of_integer_def: fixes R :: "('a, 'b) ring_scheme"   and i :: "int" shows "\<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub> = (if 0 \<le> i then \<guillemotleft>nat i\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> else \<ominus>\<^bsub>R\<^esub> \<guillemotleft>nat (- i)\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub>)"
select_convs_2: fixes initial :: "char list \<Rightarrow> nat option"   and final :: "char list \<Rightarrow> nat option"   and more :: "'a" shows "final \<lparr>initial = initial, final = final, \<dots> = more\<rparr> = final"
ide_dom: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.VVV.arr f" shows "C.VVV.ide (C.VVV.dom f)"
trans_def: fixes r :: "('g \<times> 'g) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
bs: shows "mset (bh @ b # bt) = mset (bh @ bt) + {#b#}"
axioms_1: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "total_preorder_on carrier le"
mem_list2: shows "\<forall>r\<in>set (characterize_root_list_p p). consistent_sign_vec qs2 r \<in> set (map (consistent_sign_vec qs2) (characterize_root_list_p p))"
append_Nil2: fixes xs :: "'b list" shows "xs @ [] = xs"
assms_4: fixes s :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" assumes "s \<in> set ps" shows "homogeneous_set s"
I_dnf: fixes \<phi> :: "'a fm"   and xs :: "'b list" assumes "nqfree \<phi>" shows "(\<exists>as\<in>set (dnf \<phi>). \<forall>a\<in>set as. I\<^sub>a a xs) = I \<phi> xs"
no_induced_modulus: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "primitive_dchar n \<chi>" shows "\<not> (\<exists>d<n. dcharacter.induced_modulus n \<chi> d)"
fresh_flattn: fixes a :: "Vars.var"   and S :: "stack_elem list" shows "atom a \<sharp> flattn S = atom a \<sharp> S"
Abs_freeRes_cases: fixes x :: "freeRes" obtains y :: "freeRes_Rep" where "x = Abs_freeRes y" and "y \<in> Collect freeRes_Rep_set"
GSMP_wt_subst_subset: fixes t :: "('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> GSMP (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)" shows "t \<in> GSMP M"
path_length: 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 Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "wf_prog wf_md P" shows "length Cs \<le> length P"
unit_disc_fix_f_congruent_triangles: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and 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"   and "u' \<in> unit_disc"   and "v' \<in> unit_disc"   and "w' \<in> unit_disc"   and "poincare_distance u v = poincare_distance u' v'"   and "poincare_distance v w = poincare_distance v' w'"   and "poincare_distance u w = poincare_distance u' w'" shows "\<exists>M. unit_disc_fix_f M \<and> M u = u' \<and> M v = v' \<and> M w = w'"
exists_x_in_M1: shows "\<exists>x. x \<in> M1"
fg: fixes x :: "'a" assumes "x \<in> topspace X"   and "p x \<in> Z frontier_of U" shows "f x = g x"
assms: shows "map_aux ` S \<subseteq> Field wo_aux"
right_one_inf: fixes a :: "'b" shows "(1::'b) \<sqinter> a = a"
is_BaseDim: fixes dim_ty_sem :: "'a itself \<Rightarrow> (int, sdim) dimvec" assumes "class.basedim_type dim_ty_sem" shows "is_BaseDim (dim_ty_sem TYPE('a))"
approachable_lt_le2: fixes Q :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>d>0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) = (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
Mf_def: fixes cfs :: "int \<times> int poly multiset" shows "pl.Mf cfs \<equiv> case cfs of (c, fs) \<Rightarrow> (pl.M c, image_mset pl.Mp fs)"
intros_3: shows "c_fst \<in> PrimRec1'"
to_fun_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "x \<in> carrier Zp" shows "f \<bullet> x \<in> carrier Zp"
FGModIso_idhom: shows "FGModuleIso G (\<cdot>) V (\<cdot>) (id \<down> V) V"
ET_Target_HAStates: 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" assumes "TS \<subseteq> ET ST" shows "Target TS \<subseteq> HAStates (HA ST)"
norm2_norm1_main_equality: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "(\<Sum>i = 0..<n. \<bar>f i\<bar>)\<^sup>2 = (\<Sum>i = 0..<n. f i * f i) + (\<Sum>i = 0..<n. \<Sum>j = 0..<n. if i = j then 0::'b else \<bar>f i\<bar> * \<bar>f j\<bar>)"
brec_initial_def: fixes Qi :: "('a, unit) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" shows "brec_initial Qi \<delta> \<equiv> let iq = brc_iq \<delta> in (brec_iqm \<delta>, hs_to_fifo.g_set_to_listr iq, brc_rcm_init \<delta>, hh_set_xx.g_disjoint_witness iq Qi)"
intro: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>R\<^esub> a \<in> I"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> x \<in> I" shows "ideal_axioms I R"
ltake_LNil: fixes n :: "enat" shows "ltake n LNil = LNil"
subgame_path_conforms_with_strategy: fixes V' :: "'a set"   and p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "V' \<subseteq> V"   and "path_conforms_with_strategy p P \<sigma>"   and "lset P \<subseteq> V'" shows "ParityGame.path_conforms_with_strategy (subgame V') p P \<sigma>"
msg6_KeyCryptNonce_disj: fixes B :: "agent"   and KC3 :: "nat"   and C :: "agent"   and N :: "nat"   and KC2 :: "nat"   and cardSK :: "nat"   and X :: "msg"   and Y :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt KC3 \<lbrace>Agent C, Nonce N, Key KC2, Key cardSK, X\<rbrace>, Y\<rbrace> \<in> set evs"   and "cardSK \<notin> symKeys"   and "evs \<in> set_cr" shows "Key cardSK \<in> analz (knows Spy evs) \<or> (\<forall>K. \<not> KeyCryptKey cardSK K evs) \<and> (\<forall>N. \<not> KeyCryptNonce cardSK N evs)"
spec4_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>2) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>2"
a'_def: shows "a' \<equiv> a + complex_of_real (e / 2)"
one_im_sheaf_def: shows "one_im_sheaf \<equiv> \<lambda>V. \<one>\<^bsub>f \<^sup>\<inverse> S V\<^esub>"
Push_inject2: fixes i :: "'a + nat"   and f :: "nat \<Rightarrow> 'a + nat"   and j :: "'a + nat"   and g :: "nat \<Rightarrow> 'a + nat" assumes "Push i f = Push j g" shows "f = g"
keys_permutep: fixes f :: "'b \<Rightarrow> 'b"   and m :: "'b \<Rightarrow>\<^sub>0 'c" assumes "bij f" shows "keys (permutep f m) = f -` keys m"
measure_cases: fixes x :: "'c measure" obtains \<Omega> :: "'c set" and A :: "'c set set" and \<mu> :: "'c set \<Rightarrow> ennreal" where "x = Abs_measure (\<Omega>, A, \<mu>)" and "\<forall>a\<in>- A. \<mu> a = 0" and "measure_space \<Omega> A \<mu>"
filterlim_inverse_at_infinity: shows "filterlim inverse at_infinity (at (0::'b))"
inf_sup_aci_6: fixes inf :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and sup :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and x :: "'l"   and y :: "'l"   and z :: "'l" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
fps_mult_left_const_nth: fixes c :: "'a"   and f :: "'a fps"   and n :: "nat" shows "fps_nth (fps_const c * f) n = c * fps_nth f n"
sdrop_smap: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and s :: "'e stream" shows "sdrop n (smap f s) = smap f (sdrop n s)"
assms_5: shows "(c, s) \<rightarrow>*c (c', s')"
simps: fixes a1 :: "classes1"   and a2 :: "classes1" shows "subclass1 a1 a2 = ((\<exists>c. a1 = c \<and> a2 = Object \<and> c \<noteq> Object) \<or> a1 = Employee \<and> a2 = Person \<or> a1 = Customer \<and> a2 = Person)"
avl_balL: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height r - 1 \<le> height l \<and> height l \<le> height r + 2" shows "avl (balL l a r)"
above_def: fixes L :: "(alt \<times> alt) set"   and a :: "alt"   and b :: "alt" shows "Arrow_Order.above L a b \<equiv> rem L b \<union> {(a, b)} \<union> {(x, b) |x. (x, a) \<in> L \<and> x \<noteq> b} \<union> {(b, y) |y. (a, y) \<in> L \<and> y \<noteq> b}"
upred_ares_aext: fixes a :: "'c \<Longrightarrow> 'd"   and P :: "'d upred" assumes "a \<natural> P" shows "(P \<restriction>\<^sub>p a) \<oplus>\<^sub>p a = P"
replicate_mset_Suc: fixes n :: "nat"   and x :: "'k" shows "replicate_mset (Suc n) x = add_mset x (replicate_mset n x)"
abs_trm_def: shows "abs_trm \<equiv> quot_type.abs (\<approx>) Abs_trm"
chamber_distance_def: fixes C :: "'a set"   and D :: "'a set" shows "chamber_distance C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. gallery (C # Cs @ [D]))))"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list" shows "xs \<up> m \<down> n = xs \<down> (n + m) \<up> m"
no_isolated_singularity: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and K :: "complex set" assumes "continuous_on S f"   and "f holomorphic_on S - K"   and "open S"   and "finite K" shows "f holomorphic_on S"