hcju/isabelleps huggingface dataset:
mat_diag_mult_right: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "A \<in> carrier_mat nr n" shows "A * mat_diag n f = mat nr n (\<lambda>(i, j). A $$ (i, j) * f j)"
assms_6: shows "q = 2 ^ k * 3 - 1"
const_term_def: fixes p :: "'a list" shows "const_term p = local.eval p \<zero>"
iff_1: fixes x :: "nat"   and y :: "nat" shows "(f x = f y) = (x = y)"
cao1: shows "r1.cond_action_oks s1 t \<lbrace>ta2\<rbrace>\<^bsub>c\<^esub>"
intros_6: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" assumes "\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> e s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t))"   and "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}" shows "\<turnstile>\<^sub>1 {P'} c { e' \<Down> Q'}"
simps_1: fixes uu :: "const" shows "(uu \<sqsubseteq> Any) = True"
cinsert_commute: fixes x :: "'d"   and y :: "'d"   and A :: "'d cset" shows "cinsert x (cinsert y A) = cinsert y (cinsert x A)"
antisym: fixes x :: "'b list"   and y :: "'b list" assumes "prefix x y"   and "prefix y x" shows "x = y"
specify_left: fixes m :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c nres"   and M :: "'c nres" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> f x \<le> M" shows "m \<bind> f \<le> M"
finite_Un: fixes F :: "'b set"   and G :: "'b set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
L_While_pfp: fixes c :: "com"   and b :: "bexp"   and X :: "char list set" shows "L c (L (WHILE b DO c) X) \<subseteq> L (WHILE b DO c) X"
coincidence_term: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and \<theta> :: "trm"   and I :: "interp" assumes "Vagree \<omega> \<omega>' (FVT \<theta>)" shows "term_sem I \<theta> \<omega> = term_sem I \<theta> \<omega>'"
simps_3: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and a :: "'a aexp" shows "gexp_constrains (Gt a1 a2) a = (aexp_constrains a1 a \<or> aexp_constrains a2 a)"
heads_face_cycle_set: fixes a :: "'b" assumes "a \<in> arcs G" shows "head G ` G.face_cycle_set a = tail G ` G.face_cycle_set a"
elims: fixes x :: "'b"   and xa :: "'c hp \<Rightarrow> 'b"   and xb :: "'c hp option"   and y :: "'b" assumes "lift_hp x xa xb = y" obtains c :: "'b" and f :: "'c hp \<Rightarrow> 'b" where "x = c__" and "xa = fa__" and "xb = None" and "y = c__"   | c :: "'b" and f :: "'c hp \<Rightarrow> 'b" and h :: "'c hp" where "x = c" and "xa = f" and "xb = Some h" and "y = f h"
index_mult_mat_2: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_row (A * B) = dim_row A"
type_definition_def: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "type_definition Rep Abs A \<equiv> (\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)"
is_isoI: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
assms_3: shows "Q\<^sub>1 \<inter> (I\<^sub>1 \<union> I\<^sub>2) \<noteq> {}"
wf_sees_clinit1: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and a :: "'a class"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>a\<rfloor>"   and "P \<turnstile> C sees clinit, b :  Ts\<rightarrow>T = m in D" shows "b = Static \<and> Ts = [] \<and> T = Void \<and> D = C"
nat_filtration_vimage_finite: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "finite (f ` space M)"
wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_def: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<equiv> subst_domain \<theta> \<inter> range_vars \<theta> = {} \<and> finite (subst_domain \<theta>)"
S4b_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $(S4 rmhist p \<and> ires!p \<noteq> #NotAResult) \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> (S4 rmhist p \<and> ires!p \<noteq> #NotAResult)$ \<or> S5 rmhist p$"
filterpositions2_def: fixes P :: "'b \<Rightarrow> bool"   and l :: "'b list" shows "filterpositions2 P l = concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l])"
bounded_iff: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
oalist_inv_def: fixes xs :: "('a \<times> 'b) list \<times> 'a key_order" shows "ko.oalist_inv xs = ko.oalist_inv_raw (snd xs) (fst xs)"
fold_equality: fixes z :: "'e"   and A :: "'d set"   and y :: "'e" assumes "fold_graph g z A y" shows "Finite_Set.fold g z A = y"
inject_3: fixes x3 :: "bool"   and y3 :: "bool" shows "(C x3 = C y3) = (x3 = y3)"
Suc: fixes x :: "'a" shows "ldropn n (iterates f x) = iterates f ((f ^^ n) x)"
dsumag_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "a\<Oplus>\<^bsub>I\<^esub> A = \<lparr>carrier = carr_dsumag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A\<rparr>"
sel_8: fixes x31 :: "poscheckvariant"   and x32 :: "('d, 'c) Term.term"   and x33 :: "('d, 'c) Term.term" shows "bvars\<^sub>s\<^sub>t\<^sub>p \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t = []"
chop_apps: fixes t :: "('a, 'b) tm"   and ts :: "('a, 'b) tm list" assumes "is_App t" shows "chop (apps t ts) = apps (chop t) ts"
simps_1: fixes a :: "('b, 'c) Combinators"   and l :: "'b set list" shows "FWNormalisationCore.insort a [] l = [a]"
dist_self: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.metric_space dist uniformity open" shows "dist x x = 0"
normalize_quot_normalize_right: fixes x :: "'b"   and y :: "'b" shows "normalize_quot (x, normalize y) = (fst (normalize_quot (x, y)) * unit_factor y, snd (normalize_quot (x, y)))"
induct: fixes P :: "pi \<Rightarrow> bool"   and pi :: "pi" assumes "P \<zero>"   and "\<And>name1 name2 pi. P pi \<Longrightarrow> P (name1{name2}.pi)"   and "\<And>pi. P pi \<Longrightarrow> P (\<tau>.pi)"   and "\<And>name1 name2 pi. P pi \<Longrightarrow> P (name1<name2>.pi)"   and "\<And>name1 name2 pi. P pi \<Longrightarrow> P ([name1\<frown>name2]pi)"   and "\<And>name1 name2 pi. P pi \<Longrightarrow> P ([name1\<noteq>name2]pi)"   and "\<And>pi1 pi2. \<lbrakk>P pi1; P pi2\<rbrakk> \<Longrightarrow> P (pi1 \<oplus> pi2)"   and "\<And>pi1 pi2. \<lbrakk>P pi1; P pi2\<rbrakk> \<Longrightarrow> P (pi1 \<parallel> pi2)"   and "\<And>name pi. P pi \<Longrightarrow> P (<\<nu>name>pi)"   and "\<And>pi. P pi \<Longrightarrow> P (!pi)" shows "P pi"
preprm_ext_def: fixes \<pi> :: "('b \<times> 'b) list"   and \<sigma> :: "('b \<times> 'b) list" shows "\<pi> =p \<sigma> \<equiv> \<forall>x. preprm_apply \<pi> x = preprm_apply \<sigma> x"
assms_2: fixes j :: "rat list" assumes "j \<in> set signs1" shows "length j = n1"
SD_def: fixes \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "SD \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ [c] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> \<beta> @ \<alpha> \<in> Tr"
l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf get_tag_name get_tag_name_locs"
absorb1: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" shows "max a b = a"
wti_def: fixes cert :: "'a list" shows "wti cert \<equiv> wtl_inst cert f r \<top> step"
Well_Shaped_Graph_def: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option" shows "Well_Shaped_Graph nodes inPorts outPorts vertices nodeOf edges hyps \<equiv> (Well_Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps \<and> Acyclic_Graph nodes inPorts outPorts vertices nodeOf edges hyps) \<and> Saturated_Graph nodes inPorts outPorts vertices nodeOf edges \<and> Pruned_Port_Graph nodes inPorts outPorts vertices nodeOf edges"
AA: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "r ~~ r'"   and "s ~~ s'" shows "Alt r s ~~ Alt r' s'"
th3_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "tarski_first5 C B" shows "B a b b"
hyps: shows "OnlyTwoNets p \<longrightarrow> NetsCollected2 p \<longrightarrow> NetsCollected p \<longrightarrow> noDenyAll1 p \<longrightarrow> allNetsDistinct p \<longrightarrow> separated p"
uminus_SUP: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Sup (B ` A)) = Inf ((\<lambda>x. uminus (B x)) ` A)"
oSuc_def: fixes x :: "ordinal" shows "oSuc x = oStrictLimit (\<lambda>n. x)"
aless_minus: fixes x :: "ant"   and y :: "ant" assumes "x < y" shows "- y < - x"
integral: fixes x :: "real" assumes "2 \<le> x" shows "(r has_integral r' x) {2..x}"
permutations_of_set_empty: shows "permutations_of_set {} = {[]}"
UN_binaryset_eq: fixes A :: "'b set"   and B :: "'b set" shows "\<Union> (range (binaryset A B)) = A \<union> B"
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"   and C :: "'d" 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; A\<^sub>R \<sharp>* C\<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)}"
add_frac_num: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero" shows "plus (divide x y) z = divide (plus x (times z y)) y"
converse_rtranclp_into_rtranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
ln_\<theta>_asymptotics: shows "(\<lambda>x. ln (\<theta> x) - ln x) \<in> o(\<lambda>_. 1)"
weight_while_conv_iter: fixes s :: "'a" shows "weight_spmf (fuse.while s) = (\<Squnion>n. Sigma_Algebra.measure (measure_spmf (fuse.iter n s)) {s. \<not> guard s})"
add_mset_remove_trivial_eq: fixes N :: "'c multiset"   and a :: "'c" shows "(N = add_mset a (N - {#a#})) = (a \<in># N)"
list_all2_all_nthI: fixes a :: "'a list"   and b :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
frontier_disjoint_eq: fixes S :: "'a set" shows "(frontier S \<inter> S = {}) = open S"
chain_sup_def: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph" shows "chain_sup S \<equiv> LG (\<Union>i. edges (S i)) (\<Union>i. vertices (S i))"
borel_measurable_uminus_eq: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "((\<lambda>x. - f x) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
map_add_subsumed2: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m g" shows "g ++ f = g"
conf_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "P \<turnstile> T \<le> T'" shows "P,h \<turnstile> v :\<le> T'"
absorb1: fixes b :: "'a"   and a :: "'a" assumes "b \<sqsubseteq> a" shows "ord.max (\<sqsubseteq>) a b = a"
extends_and_sp_imp_not_using_sub: fixes v :: "'c \<times> nat"   and v' :: "'c \<times> nat"   and g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and subs' :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat" assumes "extends g subs (v, v') subs'"   and "SubRel.subpath g v1 es v2 subs" shows "\<not> uses_sub v1 es v2 (v, v')"
freq_0_right: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and a :: "'a" assumes "alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}"   and "a \<in> alphabet t\<^sub>1" shows "freq t\<^sub>2 a = 0"
insert_if: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a measure"   and x :: "'b" assumes "finite A" shows "sup_measure.F g (insert x A) = (if x \<in> A then sup_measure.F g A else sup (g x) (sup_measure.F g A))"
PS: fixes i :: "nat" assumes "P i" shows "\<exists>j>i. P j \<and> Q j"
lfp_def: 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 f :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.lfp Inf less_eq f = Inf {u. less_eq (f u) u}"
Ball_set: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
dom_cod: fixes f :: "'b" assumes "J1.arr f" shows "J1.dom (J1.cod f) = J1.cod f"
abs_eq: fixes x :: "'a multiset \<times> 'a multiset" shows "zcount (abs_zmultiset x) = (case x of (Mp, Mn) \<Rightarrow> \<lambda>x. int (count Mp x) - int (count Mn x))"
single_not_bot: fixes x :: "'a" shows "Predicate.single x \<noteq> \<bottom>"
ideal_inc_set_multTr: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and n :: "nat" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "ideal R C"   and "set_mult R A B \<subseteq> C" shows "\<forall>f\<in>{j. j \<le> n} \<rightarrow> set_mult R A B. \<Sigma>\<^sub>e R f n \<in> C"
parts_synth: fixes H :: "msg set" shows "parts (synth H) = parts H \<union> synth H"
int_simps_30: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<forall>x. P) = P"
finitePairSecondRange: fixes R :: "('c \<times> 'd set) set" shows "(\<forall>pair\<in>R. finite (snd pair)) = (\<forall>y\<in>Range R. finite y)"
greatest_le: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set"   and a :: "'a" assumes "greatest L x A"   and "a \<in> A" shows "a \<sqsubseteq>\<^bsub>L\<^esub> x"
norm_Pair1: fixes x :: "'b" shows "norm (0::'a, x) = norm x"
ssaStep_def: fixes g :: "'g"   and m :: "'node"   and i :: "nat"   and s :: "'val \<Rightarrow> 'node option"   and v :: "'val" shows "ssaStep g m i s v \<equiv> if v \<in> defs g m then Some m else case phis g (m, v) of None \<Rightarrow> s v | Some phiParams \<Rightarrow> s (phiParams ! i)"
chebyshev_prob_zero: fixes Y :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>y. (Y y)\<^sup>2)"   and "random_variable borel Y"   and "0 < expectation Y" shows "prob {a \<in> space M. Y a = 0} \<le> expectation (\<lambda>y. (Y y)\<^sup>2) / (expectation Y)\<^sup>2 - 1"
assms_1: shows "SubnetS s t \<in> reachable (closed (pnet np (p1 \<parallel> p2))) I"
pk: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> L a"   and "(s, u) \<in> R b" shows "\<exists>xs. local.lpeak r a b xs \<and> (t, u) \<in> local.lconv xs"
Rats_cases: fixes q :: "'b" assumes "q \<in> \<rat>" obtains r :: "rat" where "q = of_rat r"
multrow_mat: fixes A :: "'c mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and a :: "'c" assumes "A \<in> carrier_mat n nc" shows "multrow k a A = multrow_mat n k a * A"
IH: shows "local.newton_coefficients_main (rev (map local.f [0..<Suc k])) (rev (map local.x [0..<Suc k])) = rev (map (\<lambda>i. map (\<lambda>j. local.xij_f j i) [0..<Suc i]) [0..<Suc k])"
cone_\<chi>: shows "\<chi> a \<in> \<chi>a.D.cones (l a)"
mC_ZOC_eff_cont_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC_eff_cont theta c d s t I0 P F \<equiv> \<forall>I i j. I \<in> P - {I0} \<and> i \<in> I \<and> j \<in> F I \<longrightarrow> eff c s i \<approx> eff d t j \<and> (local.cont c s i, local.cont d t j) \<in> theta"
fmrel_fmdom_eq: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'f \<rightharpoonup>\<^sub>f 'd"   and y :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "fmrel P x y" shows "fmdom x = fmdom y"
P1: shows "Bet A CC D \<and> (CC = B \<or> B Out CC C)"
path_connected_imp_connected_space: fixes X :: "'b topology" assumes "path_connected_space X" shows "connected_space X"
restrict_to_halfspaces_clw_def: shows "restrict_to_halfspaces_clw = restrict_to_halfspaces"
image_cong: fixes M :: "'f set"   and N :: "'f set"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
nth_default_strip_while_dflt: fixes dflt :: "'f"   and xs :: "'f list" shows "nth_default dflt (strip_while ((=) dflt) xs) = nth_default dflt xs"
weak_sharing_consistent_append: fixes \<O> :: "nat set"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "weak_sharing_consistent \<O> (xs @ ys) = (weak_sharing_consistent \<O> xs \<and> weak_sharing_consistent (acquired True xs \<O>) ys)"
gt_sym_antisym: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and g :: "'a" assumes "gt_sym gt_sym"   and "gt_sym f g" shows "\<not> gt_sym g f"
\<alpha>_def: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "subresultant_prs_locale.\<alpha> \<delta> f i = f (i - 1) ^ Suc (\<delta> (i - 2))"
fst2: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A >>> A'" shows "Fst (SimplyTyped.Pair A B) >>> A'"
union_comm: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = B \<union> A"
Restrict_subset: fixes A :: "'d set"   and r :: "('d \<times> 'e) set" shows "Restrict A r \<subseteq> r"
jump_jump_poly_aux: fixes p :: "real poly"   and q :: "real poly"   and a :: "real" assumes "p \<noteq> 0"   and "coprime p q" shows "jump (\<lambda>x. poly q x / poly p x) a = jump_poly q p a"
step: shows "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
euler_60_decimals: shows "\<bar>exp 1 - 2718281828459045235360287471352662497757247093699959574966968 / 10 ^ 60\<bar> \<le> inverse (10 ^ 60)"
reduced_homology_group_nsphere_1: fixes n :: "nat" shows "reduced_homology_group (int n) (nsphere n) \<cong> integer_group"
subst3: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'a"   and x :: "name" assumes "length xvec = length Tvec"   and "distinct xvec"   and "set xvec \<subseteq> supp T"   and "x \<sharp> T[xvec::=Tvec]" shows "x \<sharp> Tvec"
prove_def: fixes s :: "(nat \<times> form) list" shows "prove s = prove' [s]"
eq_replicate_iff: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" shows "(xs = replicate n x) = (set xs \<subseteq> {x} \<and> length xs = n)"
non_empty_ith_2: fixes C :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" assumes "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "ith (eq_upto C (lexmin C) i) i \<noteq> {}"
nomulti_digraph_axioms: shows "nomulti_digraph (app_iso hom G)"
left_options_def: fixes g :: "game" shows "left_options g \<equiv> zimage Abs_game (zexplode (Fst (Rep_game g)))"
rel_gpv''_lossless_gpvD1: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I> :: "('i, 'k) \<I>"   and \<I>' :: "('j, 'l) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "lossless_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "lossless_gpv \<I>' gpv'"
WT_calleeD: fixes \<I> :: "('h, 'i) \<I>"   and callee :: "'h \<Rightarrow> ('i \<times> 'j) spmf"   and ret :: "'i"   and s :: "'j"   and out :: "'h" assumes "\<I> \<turnstile>c callee \<surd>"   and "(ret, s) \<in> set_spmf (callee out)"   and "out \<in> outs_\<I> \<I>" shows "ret \<in> responses_\<I> \<I> out"
intros_2: fixes start :: "'a option"   and list :: "('a \<times> 'a option) list"   and prev :: "'a"   and ref :: "'a option"   and oid :: "'a" assumes "insert_seq start (list @ [(prev, ref)])" shows "insert_seq start (list @ [(prev, ref), (oid, Some prev)])"
is_rcoset_replist_imp_nrelated_nth: fixes H :: "'g set"   and gs :: "'g list"   and i :: "nat"   and j :: "nat" assumes "Supgroup.Subgroup H"   and "Supgroup.is_rcoset_replist H gs"   and "i < length gs"   and "j < length gs"   and "i \<noteq> j" shows "gs ! i - gs ! j \<notin> H"
add_comm': fixes a :: "int"   and b :: "int"   and p\<^sub>1 :: "int point"   and p\<^sub>2 :: "int point"   and p\<^sub>3 :: "int point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3" shows "local.add a p\<^sub>2 (local.add a p\<^sub>1 p\<^sub>3) = local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3)"
generating_ring: shows "ring_of_sets \<Omega> A.generated_ring"