hcju/isabelleps huggingface dataset:
update_convs_2: fixes lowlink' :: "('c \<Rightarrow> nat option) \<Rightarrow> 'c \<Rightarrow> nat option"   and counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "lowlink_update lowlink' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink' lowlink, tj_stack = tj_stack, \<dots> = more\<rparr>"
mult_ge1_mono_1: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "(1::'c) \<le> b" shows "a \<le> a * b"
Equality: fixes a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
word_set_nth: fixes w :: "'b word"   and n :: "nat" shows "set_bit_class.set_bit w n (w !! n) = w"
dghm_comp_is_fl_dghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
that_1: assumes "ord a b = LESS"   and "a < b" shows "thesis"
vector_eq_dot_span: fixes x :: "'c"   and B :: "'c set"   and y :: "'c" assumes "x \<in> span B"   and "y \<in> span B"   and "\<And>i. i \<in> B \<Longrightarrow> i \<bullet> x = i \<bullet> y" shows "x = y"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* C" shows "Prop (P' \<parallel> Q)"
finsum_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> nat \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier P"   and "f a \<in> carrier P" shows "finsum P f (insert a F) = f a \<oplus>\<^bsub>P\<^esub> finsum P f F"
aND_sort: fixes p :: "('a set set, 'b) Combinators list"   and l :: "'a set set set list" assumes "singleCombinators p"   and "allNetsDistinct p" shows "allNetsDistinct (FWNormalisationCore.sort p l)"
one_smult: fixes x :: "('c, 'd) sq_matrix" shows "(1::'c) *\<^sub>S x = x"
transpose_transpose: fixes A :: "('c, 'd) sq_matrix" shows "Square_Matrix.transpose (Square_Matrix.transpose A) = A"
simps_5: fixes x21 :: "dB"   and x22 :: "dB"   and x1 :: "nat" shows "x21 \<degree> x22 \<noteq> Var x1"
iprev_inext: fixes n :: "nat"   and I :: "nat set" assumes "n \<noteq> Max I \<or> infinite I" shows "iprev (inext n I) I = n"
map_add_ac_2: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>1 ++ h\<^sub>0"
finite_comm_group_Characters: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_comm_group G" shows "finite_comm_group (Characters G)"
slength_strict: shows "slength\<cdot>\<bottom> = \<bottom>"
mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
rel_st_def: fixes r :: "'a \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and r' :: "'b \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_st r r' = (\<lambda>(u, v) (u', v') x z. inpt_st r u u' x \<and> (\<forall>y. r u u' x y \<longrightarrow> inpt_st r' v v' y) \<and> (r u u' OO r' v v') x z)"
meas: fixes F :: "'b \<Rightarrow> ennreal" assumes "F \<in> borel_measurable N" shows "f F \<in> borel_measurable N"
jkbpC_tLength_inv: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> jkbpC"   and "tLength t = n" shows "t \<in> jkbpC\<^bsub>n\<^esub>"
nonzero_mult_in_car: fixes a :: "'a"   and b :: "'a" assumes "a \<in> nonzero R"   and "b \<in> nonzero R" shows "a \<otimes> b \<in> carrier R"
starfun_pow3: fixes n :: "nat"   and R :: "'b star" shows "(*f* (\<lambda>r. r ^ n)) R = R pow hypnat_of_nat n"
lappend_code_2: fixes xa :: "'b"   and x :: "'b llist"   and ys :: "'b llist" shows "(xa ## x) @@ ys = xa ## x @@ ys"
spmf_of_set_def: fixes A :: "'b set" shows "spmf_of_set A = (if finite' A then spmf_of_pmf (pmf_of_set A) else return_pmf None)"
Int_iff: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
oo: fixes U :: "'a set" assumes "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
e_def: shows "e = exp (- epsilonG TYPE('a) * M0) / 2"
pow_non_prim: fixes k :: "nat"   and w :: "'b list" assumes "1 < k" shows "\<not> primitive (w \<^sup>@ k)"
nf_RRR_8: shows "test (C \<circ> K \<circ> C \<circ> K) = [True, False, False, False, True]"
induct: fixes P :: "'a formula \<Rightarrow> bool"   and a0 :: "'a formula" assumes "\<And>k. P (Atom k)"   and "P \<bottom>"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<and> G))"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<or> G))"   and "\<And>F. P F \<Longrightarrow> P (\<^bold>\<not> (\<^bold>\<not> F))"   and "\<And>F G. \<lbrakk>P F; P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<rightarrow> G))"   and "\<And>v. P (\<^bold>\<not> (Atom v))"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G)" shows "P a0"
zassenhaus_reconstruction_irreducible\<^sub>d: fixes fs :: "int poly list" assumes "zassenhaus_reconstruction hs p n f = fs" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
restrict_idle: fixes n :: "nat"   and A :: "nat set"   and v :: "'c option list" assumes "wf_tuple n A v" shows "restrict A v = v"
hyps: assumes "valid_unMultigraph g" shows "valid_unMultigraph (rem_unPath xs g)"
Real_induct: fixes P :: "real \<Rightarrow> bool"   and x :: "real" assumes "\<And>X. Real.cauchy X \<Longrightarrow> P (Real.Real X)" shows "P x"
interior_of_restrict: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S = X interior_of (topspace X \<inter> S)"
Brouwer_degree2_section_map: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "section_map (nsphere p) (nsphere p) f" shows "\<bar>Brouwer_degree2 p f\<bar> = 1"
span_zero: fixes S :: "'b set" shows "(0::'b) \<in> local.span S"
mult_imp_less_div_pos: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less (times z y) x" shows "less z (divide x y)"
index_add_mat_2: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_row (A + B) = dim_row B"
col_of_dagger: fixes j :: "nat"   and M :: "complex mat" assumes "j < dim_row M" shows "col M\<^sup>\<dagger> j = vec (dim_col M) (\<lambda>i. cnj (M $$ (j, i)))"
indRelTEQ_impl_TRel_is_weak_reduction_correspondence_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_correspondence_simulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_correspondence_simulation (TRel\<^sup>*) Target"
select_convs_16: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = sip"
sub: shows "keys (monomial (1::'a) x) \<subseteq> {x}"
setops\<^sub>s\<^sub>s\<^sub>t_are_pairs: fixes t :: "('fun, 'a) Term.term"   and A :: "('fun, 'a) stateful_strand_step list" assumes "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t A" shows "\<exists>s s'. t = pair (s, s')"
LPDi_Suc_alt: fixes i :: "nat"   and r :: "mregex" shows "LPDi (Suc i) r = \<Union> (LPD ` LPDi i r)"
assms_6: shows "\<forall>j. xs !! j \<in> V"
iFROM_set_UN_set_eq: shows "iFROM_set = iFROM_UN_set"
isPrefixPrefixBeforeLastMarked: fixes M :: "('b \<times> bool) list" shows "isPrefix (prefixBeforeLastMarked M) M"
IH: fixes f :: "dB" shows "norm_pn f n \<simeq> Pure f \<diamondop> n"
lemma1: fixes B :: "'c set"   and A2 :: "'c set"   and F :: "'c program"   and A1 :: "'c set"   and C :: "'c set" assumes "B \<subseteq> A2"   and "F \<in> A1 - B co A1 \<union> B"   and "F \<in> A2 - C co A2 \<union> C" shows "F \<in> A1 \<union> A2 - C co A1 \<union> A2 \<union> C"
MP_same: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IMP B"   and "H \<turnstile> A" shows "H \<turnstile> B"
rep_eq: fixes x :: "'c"   and xa :: "'d"   and xb :: "('c, 'd) fmap" shows "fmlookup (fmupd x xa xb) = map_upd x xa (fmlookup xb)"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F F \<epsilon>FoF\<eta>.map"
power2_norm_eq_inner: fixes x :: "'a" shows "(norm x)\<^sup>2 = inner x x"
fds_nth_mult: fixes f :: "'a fds"   and g :: "'a fds" shows "fds_nth (f * g) = dirichlet_prod (fds_nth f) (fds_nth g)"
I_lift_nnf_qe_normal: fixes qe :: "'a fm \<Rightarrow> 'a fm"   and \<phi> :: "'a fm"   and xs :: "'b list" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>\<phi>. \<lbrakk>nqfree \<phi>; normal \<phi>\<rbrakk> \<Longrightarrow> normal (qe \<phi>)"   and "\<And>xs \<phi>. \<lbrakk>normal \<phi>; nqfree \<phi>\<rbrakk> \<Longrightarrow> I (qe \<phi>) xs = (\<exists>x. I \<phi> (x # xs))"   and "normal \<phi>" shows "I (lift_nnf_qe qe \<phi>) xs = I \<phi> xs"
cSUP_UNION: fixes A :: "V set"   and f :: "V \<Rightarrow> V"   and B :: "V \<Rightarrow> V set" assumes "small A"   and "small (\<Union>x\<in>A. f ` B x)" shows "\<Squnion> (f ` \<Union> (B ` A)) = (SUP x\<in>A. \<Squnion> (f ` B x))"
insert_map_eq_diff: fixes i :: "nat"   and s :: "'c"   and f :: "nat \<Rightarrow> 'c"   and j :: "nat"   and t :: "'c"   and g :: "nat \<Rightarrow> 'c"   and s' :: "'c" assumes "insert_map i s f = insert_map j t g"   and "i \<noteq> j" shows "\<exists>g'. insert_map i s' f = insert_map j t g'"
set_subset_Univ: fixes a :: "'t" assumes "S'.ide a" shows "S'.set a \<subseteq> S'.Univ"
connectedE: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "Graph.connected G v w" obtains xs :: "'c list" where "Graph.walk G xs" and "xs \<noteq> []" and "hd xs = v" and "last xs = w"
hypnat_infinite_has_nonstandard: fixes A :: "nat set" assumes "infinite A" shows "hypnat_of_nat ` A \<subset> *s* A"
inject: fixes x21 :: "'b"   and x22 :: "'b list"   and y21 :: "'b"   and y22 :: "'b list" shows "(Cons x21 x22 = Cons y21 y22) = (x21 = y21 \<and> x22 = y22)"
decseq_def: fixes X :: "nat \<Rightarrow> 'a" shows "decseq X = (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
eq: shows "snd b \<oplus> (snd c \<oplus> fst a) = snd a \<oplus> (snd c \<oplus> fst b)"
mono_quorum: fixes Q :: "process set"   and Q' :: "process set" assumes "Q \<in> Quorum"   and "Q \<subseteq> Q'" shows "Q' \<in> Quorum"
Iio_Int_singleton: fixes k :: "'a"   and x :: "'a" shows "{..<k} \<inter> {x} = (if x < k then {x} else {})"
power_Suc2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc n) = times (power.power one times a n) a"
top_conj_1: fixes x :: "'c"   and P :: "bool" shows "(\<top> x \<and> P) = P"
eq_impD: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and s :: "'b"   and s' :: "'b" assumes "eq_imp f g"   and "\<forall>x. f x s = f x s'" shows "g s = g s'"
elims: fixes x :: "com"   and xa :: "char list set"   and y :: "com" assumes "bury x xa = y" obtains X :: "char list set" where "x = SKIP" and "xa = X__" and "y = SKIP"   | xb :: "char list" and a :: "aexp" and X :: "char list set" where "x = xb ::= a" and "xa = Xa__" and "y = (if xb \<in> Xa__ then xb ::= a else SKIP)"   | c\<^sub>1 :: "com" and c\<^sub>2 :: "com" and X :: "char list set" where "x = c\<^sub>1__;; c\<^sub>2__" and "xa = Xb__" and "y = bury c\<^sub>1__ (L c\<^sub>2__ Xb__);; bury c\<^sub>2__ Xb__"   | b :: "bexp" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" and X :: "char list set" where "x = IF b__ THEN c\<^sub>1 ELSE c\<^sub>2" and "xa = Xc__" and "y = IF b__ THEN bury c\<^sub>1 Xc__ ELSE bury c\<^sub>2 Xc__"   | b :: "bexp" and c :: "com" and X :: "char list set" where "x = WHILE b DO c" and "xa = X" and "y = WHILE b DO bury c (L (WHILE b DO c) X)"
p\<^sub>1_simps: shows "trg g\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g)"
finite_divisor_set: fixes n :: "nat" shows "finite (divisor_set n)"
get_host_locs_def: shows "get_host_locs \<equiv> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host_locs get_shadow_root_locs"
bell_11_index_2: shows "|\<beta>\<^sub>1\<^sub>1\<rangle> $$ (1, 0) = complex_of_real (1 / sqrt 2)"
does_eval: shows "eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C"
monom_inv_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c \<times> nat) list" assumes "class.linorder less_eq less" shows "linorder.monom_inv less_eq m \<equiv> (\<forall>(x, n)\<in>set m. 1 \<le> n) \<and> distinct (map fst m) \<and> linorder.sorted less_eq (map fst m)"
lcd': shows "l' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k'"
tr: shows "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans (opaodv i)"
stail_append2: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "xs \<noteq> [::]" shows "stail\<cdot>(xs :@ ys) = stail\<cdot>xs :@ ys"
inv\<^sub>3_def: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L = (inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L \<and> B\<^sub>2 \<subseteq> {u \<in> U. w u \<le> real c / 2})"
p_dist_sup: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) = - x \<squnion> - y"
nth_default_append: fixes dflt :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and n :: "nat" shows "nth_default dflt (xs @ ys) n = (if n < length xs then xs ! n else nth_default dflt ys (n - length xs))"
meet_bot1: fixes y :: "'c" shows "\<bottom> \<sqinter> y = \<bottom>"
sorted_merge_all: fixes xss :: "'c list list" assumes "\<forall>xs\<in>set xss. sorted xs" shows "sorted (merge_all xss)"
simps_20: shows "OUTfromCh data20 = {data17, data22}"
semiring_1_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "class.semiring_1 one times plus zero \<equiv> class.monoid_mult one times \<and> class.semiring_0 plus zero times \<and> class.zero_neq_one one zero"
postcomputation: fixes p :: "'a"   and y :: "'a"   and x :: "'a" assumes "test p"   and "pres y (!p)" shows "!p \<cdot> y + p \<cdot> (p \<cdot> x \<cdot> (!p \<cdot> y + p))\<^sup>\<dagger> \<cdot> !p = (p \<cdot> x)\<^sup>\<dagger> \<cdot> !p \<cdot> y"
simps_1: fixes x :: "'g"   and ya :: "'g" shows "(object_ptr.Ext x = object_ptr.Ext ya) = (x = ya)"
mult_neg_pos: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "(0::'a) < b" shows "a * b < (0::'a)"
euclidean_size_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size one = 1"
mydiv_def: fixes a :: "nat"   and k :: "nat" shows "mydiv a k = (if k dvd a then a div k else a div k + 1)"
filter_False: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
path_weight_refl: fixes loc :: "'loc" shows "(0::'sum) \<in>\<^sub>A path_summary loc loc"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<chi> f = C.null"
vec_first_append: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n" shows "vec_first (v @\<^sub>v w) n = v"
dickson_less_vD2: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t"   and u :: "'t" assumes "dickson_less_v d m v u" shows "d (pp_of_term u) \<le> m"
Deriv_def: fixes x :: "'c"   and A :: "'c list set" shows "Deriv x A = {xs. x # xs \<in> A}"
FactGroup_universal: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme"   and N :: "'b set" assumes "h \<in> hom G H"   and "N \<lhd> G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G; N #>\<^bsub>G\<^esub> x = N #>\<^bsub>G\<^esub> y\<rbrakk> \<Longrightarrow> h x = h y" obtains g :: "'b set \<Rightarrow> 'c" where "g \<in> hom (G Mod N) H" and "\<And>x. x \<in> carrier G \<Longrightarrow> g (N #>\<^bsub>G\<^esub> x) = h x"
hbaseI2: fixes i :: "nat"   and b :: "nat"   and n :: "C"   and m :: "C list" assumes "i < b"   and "n \<in> hbase b"   and "C m \<in> hbase b"   and "\<And>m'. m' \<in> set m \<Longrightarrow> evalC b n < evalC b m'" shows "C (replicate i n @ m) \<in> hbase b"
fds_abs_converges_mult: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_abs_converges f s"   and "fds_abs_converges g s" shows "fds_abs_converges (f * g) s"
s1: fixes s2 :: "nat"   and x :: "nat"   and y :: "nat"   and s1 :: "nat" assumes "s2 = (x * y + q - s1) mod q"   and "x < q"   and "y < q"   and "s1 < q" shows "s1 = (x * y + q - s2) mod q"
rel_fun_eq_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
exteq_refl: fixes f :: "recf" shows "f \<simeq> f"
independent_card_le_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "vs1.independent B" shows "card B \<le> vs1.dim V"
part_equivp_refl_symp_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "part_equivp R = ((\<exists>x. R x x) \<and> symp R \<and> transp R)"
set_split': fixes L :: "atomUni list"   and eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list"   and P :: "atomUni \<Rightarrow> 'a" assumes "separateAtoms L = (eq, les, leq, neq)" shows "set (map P L) = set (map (P \<circ> EqUni) eq @ map (P \<circ> LessUni) les @ map (P \<circ> LeqUni) leq @ map (P \<circ> NeqUni) neq)"
eOp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and einp :: "'index \<Rightarrow> 'c withERR option"   and ebinp :: "'bindex \<Rightarrow> 'd withERR option"   and X :: "'c" assumes "eOp MOD delta einp ebinp = OK X" shows "\<exists>inp binp. einp = OKI inp \<and> ebinp = OKI binp \<and> X = igOp MOD delta inp binp \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp"
fine_Un: fixes d :: "'a \<Rightarrow> 'b set"   and p1 :: "('a \<times> 'b set) set"   and p2 :: "('a \<times> 'b set) set" assumes "d fine p1"   and "d fine p2" shows "d fine p1 \<union> p2"
h'_simps: fixes x :: "'v"   and y :: "'v" shows "h' (x, y) = (if edge \<Gamma> x y then h 0 (x, y) + g (\<langle>y\<rangle>, \<langle>x\<rangle>) - g (\<langle>x\<rangle>, \<langle>y\<rangle>) else 0)"
connected_punctured_universe: fixes a :: "'b" assumes "2 \<le> DIM('b)" shows "connected (- {a})"
intro: fixes s :: "nat \<Rightarrow> nat option" assumes "s \<in> \<R>" shows "cons_lim s"
intros_1: fixes S :: "'b set" shows "(0::'b) \<in> \<langle>S\<rangle>"
elims_2: fixes x :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and xa :: "('f, 'i, 'j) new_thread_action list" assumes "thread_oks x xa" obtains ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" where "x = tsa__" and "xa = []"   | ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" and ta :: "('f, 'i, 'j) new_thread_action" and tas :: "('f, 'i, 'j) new_thread_action list" where "x = ts" and "xa = ta # tas" and "thread_ok ts ta \<and> thread_oks (redT_updT' ts ta) tas"
log2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and op0 :: "lop"   and e2 :: "exp0"   and bv :: "v" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_log op0 v1 e2 = Some (Val bv)" shows "evaluate ck env s1 (Log op0 e1 e2) (s2, Rval bv)"
vec_of_basis_enum_cspan: fixes X :: "'b set" assumes "length canonical_basis = d" shows "vec_of_basis_enum ` cspan X = local.span (vec_of_basis_enum ` X)"
ennreal_neq_top: fixes r :: "real" shows "ennreal r \<noteq> top_class.top"
FOREACH_rule_map: fixes g :: "'g \<Rightarrow> 'h option"   and I :: "('g \<Rightarrow> 'h option) \<Rightarrow> 'i \<Rightarrow> bool"   and s :: "'i"   and P :: "'i \<Rightarrow> bool"   and f :: "'g \<times> 'h \<Rightarrow> 'i \<Rightarrow> 'i nres" assumes "finite (dom g)"   and "I Map.empty s"   and "\<And>s. I g s \<Longrightarrow> P s"   and "\<And>h k v s. \<lbrakk>h \<subseteq>\<^sub>m g; I h s; g k = Some v; k \<notin> dom h\<rbrakk> \<Longrightarrow> f (k, v) s \<le> SPEC (I (h(k \<mapsto> v)))" shows "FOREACH (map_to_set g) f s \<le> SPEC P"
order_refl: fixes x :: "'c list" shows "x \<le> x"