hcju/isabelleps huggingface dataset:
rank_mat_mul_right: fixes A :: "'a mat"   and d :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n d"   and "B \<in> carrier_mat d nc" shows "rank (A * B) \<le> rank A"
Collect_cong: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
cloT: shows "closedin (top_of_set (U - S)) T"
dvd: shows "poly_mod.dvdm (p ^ l) u (smult c f)"
monom_mult_zero_left: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair (0::'d) t p = 0"
trans: fixes b :: "'f"   and a :: "'f"   and c :: "'f" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
\<O>'_def: fixes s :: "'a" shows "\<O>' s \<equiv> case map_of (map (\<lambda>(s, \<mu>). (s, pmf_of_alist \<mu>)) \<O>) s of None \<Rightarrow> return_pmf (hd \<O>\<^sub>s) | Some s \<Rightarrow> s"
\<omega>_limit_set_in_compact_connected: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "connected (rev.\<omega>_limit_set x)"
ProcSpec: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'f" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) Call p (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return c Q,A"
has_sort_Ty: fixes tcs :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option"   and \<kappa> :: "String.literal"   and dm :: "String.literal \<Rightarrow> String.literal set list option"   and S :: "String.literal set"   and sub :: "(String.literal \<times> String.literal) set"   and Ts :: "typ list" assumes "tcs \<kappa> = Some dm"   and "\<forall>c\<in>S. \<exists>Ss. dm c = Some Ss \<and> list_all2 (has_sort (sub, tcs)) Ts Ss" shows "has_sort (sub, tcs) (Ty \<kappa> Ts) S"
Skip: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (SKIP, s) \<Rightarrow> s"
insert_rank_invar: fixes q :: "('c, 'd) SkewBinomialTree list"   and e :: "'c"   and a :: "'d" assumes "rank_skew_invar q" shows "rank_skew_invar (local.insert e a q)"
assms_4: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> ma"
rev_hull1: fixes x :: "'b list"   and G :: "'b list set" assumes "x \<in> \<langle>rev ` G\<rangle>" shows "x \<in> rev ` \<langle>G\<rangle>"
write0_cont: fixes P :: "'a \<Rightarrow> 'b process"   and a :: "'b" assumes "cont P" shows "cont (\<lambda>x. a \<rightarrow> P x)"
exhaust: fixes y :: "'d + 'e" obtains x1 :: "'d" where "y = Inl x1"   | x2 :: "'e" where "y = Inr x2"
comp_null_1: fixes f :: "'a \<times> 'a" shows "B.VV.comp B.VV.null f = B.VV.null"
1: shows "nth_least {i \<in> s. enat i < llength w} ` {i \<in> t. enat i < llength (lselect s w)} = {i \<in> nth_least s ` t. enat i < llength w}"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>"
connect_def: fixes d :: "('y, 'z) distinguisher"   and res :: "('y, 'z) resource" shows "connect d res = run_gpv run_resource d res"
gState_progress_rel_channels_update: fixes prog :: "program"   and g :: "'b gState_scheme"   and c :: "channel"   and i :: "nat" assumes "gState_inv prog g"   and "channel_inv c"   and "i < length (gState.channels g)" shows "(g, gState.channels_update (\<lambda>cs. cs[i := c]) g) \<in> gState_progress_rel prog"
ball_simps_9: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>f ` A. P x) = (\<forall>x\<in>A. P (f x))"
rep_eq: fixes x :: "('c, 'd) mapping_rbt" shows "RBT_Mapping2.init x = rbt_init (mapping_rbt.impl_of x)"
simps_13: fixes sid' :: "nat \<Rightarrow> nat"   and id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "sid_update sid' \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid' sid, \<dots> = more\<rparr>"
bind_altc1: fixes C :: "'c cset"   and g :: "'c \<Rightarrow> 'm"   and f :: "'a \<Rightarrow> 'm" shows "bind (altc C g) f = altc C (\<lambda>c. bind (g c) f)"
simps_4: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a"   and x1 :: "int" shows "(case Const x1 of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = f1 x1"
hoareQ_sound_complete: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "(\<turnstile>\<^sub>2\<^sub>' {P} c {Q}) = (\<Turnstile>\<^sub>2\<^sub>' {P} c {Q})"
fmdrop_fset_null: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdrop_fset {||} m = m"
OpCatOpCat: fixes C :: "('d, 'e, 'f) Category_scheme" shows "Op Op C = C"
A_5: shows "\<R> = {region X I r |I r. valid_region X k I r}"
disj: shows "0 < Inf_aform' p X \<or> Sup_aform' p X < 0"
dynmethd_def: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname" shows "dynmethd G statC dynC = (\<lambda>sig. if G\<turnstile>dynC\<preceq>\<^sub>C statC then case methd G statC sig of None \<Rightarrow> None | Some statM \<Rightarrow> class_rec G dynC Map.empty (\<lambda>C c subcls_mthds. subcls_mthds ++ filter_tab (\<lambda>_ dynM. G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM \<or> dynM = statM) (methd G C)) sig else None)"
Pd'_d'_linear: shows "Linear_order (Pd' (Xd x))"
Legendre_eq_0_iff: fixes p :: "int"   and a :: "int" assumes "prime p" shows "(Legendre a p = 0) = (\<not> coprime a p)"
sup_left_divisibility: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<exists>z. x \<squnion> z = y)"
abs_summable_on_reindex_bij_betw: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "bij_betw g A B" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on B)"
simps_2: fixes i :: "int"   and j :: "int"   and l :: "int tree"   and k :: "int"   and r :: "int tree" shows "local.wpl i j \<langle>l, k, r\<rangle> = local.wpl i (k - 1) l + local.wpl (k + 1) j r + w i j"
ftif_cons_trueI: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and ty :: "ty"   and f :: "char list"   and fd_list :: "fd list" assumes "find_type P ctx cl (Some ty)" shows "ftype_in_fds P ctx ([fd_def cl f] @ fd_list) f (ty_opt_bot_opt (Some ty))"
matrix_matrix_mult_def: fixes m :: "(('e, 'h) vec, 'g) vec"   and m' :: "(('e, 'f) vec, 'h) vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $ i $ k * m' $ k $ j"
closedin_path_component_locally_path_connected: fixes S :: "'b set"   and x :: "'b" assumes "locally path_connected S" shows "closedin (top_of_set S) (path_component_set S x)"
infsetsum_cmult_left: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "c \<noteq> (0::'b) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. f x * c) = infsetsum f A * c"
summable_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "summable (\<lambda>n. f (Suc n)) = summable f"
unanimity: fixes P :: "indi \<Rightarrow> alt \<Rightarrow> real"   and a :: "alt"   and b :: "alt" assumes "\<And>i. P i a < P i b" shows "F P a < F P b"
induct_step: fixes st :: "'a" shows "\<lbrace>b \<and> p \<and> e =\<^sub>u \<guillemotleft>st\<guillemotright>\<rbrace> Q \<lbrace>p \<and> (e, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright>\<rbrace>\<^sub>u"
Bird_strategy: fixes pat :: "[:'r:]"   and p :: "'s \<rightarrow> tr"   and op :: "'s \<rightarrow> 'r \<rightarrow> 's"   and z :: "'s"   and step :: "Integer \<times> 's \<rightarrow> 'r \<rightarrow> Integer \<times> 's" assumes "endswith\<cdot>pat = p oo sfoldl\<cdot>op\<cdot>z"   and "step = (\<Lambda> (n, x) y. (n + 1, op\<cdot>x\<cdot>y))"   and "p\<cdot>\<bottom> = \<bottom>" shows "matches\<cdot>pat = smap\<cdot>cfst oo sfilter\<cdot>(p oo csnd) oo sscanl\<cdot>step\<cdot>(0, z)"
unch: shows "|~ Unchanged (ic, q, oc) \<longrightarrow> BInv ic q oc \<longrightarrow> \<circle>BInv ic q oc"
less_imp_erfc_real_less: fixes a :: "real"   and b :: "real" assumes "a < b" shows "erfc b < erfc a"
subfrmlsn_card: fixes \<phi> :: "'b ltln" shows "card (subfrmlsn \<phi>) \<le> size \<phi>"
sort_oalist_raw_id: fixes xs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs" shows "sort_oalist_raw xs = xs"
dist_triangle_half_l: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x1 :: "'b"   and y :: "'b"   and e :: "real"   and x2 :: "'b" assumes "class.metric_space dist uniformity open"   and "dist x1 y < e / 2"   and "dist x2 y < e / 2" shows "dist x1 x2 < e"
ipurge_ref_all: fixes D :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and xs :: "'c list"   and u :: "'b"   and I :: "('b \<times> 'b) set" assumes "\<not> (\<exists>v\<in>D ` (X \<union> set xs). (u, v) \<in> I)" shows "ipurge_ref I D u xs X = X"
to_bl_0: shows "to_bl 0 = replicate LENGTH('b) False"
Always_eq_includes_reachable: fixes A :: "'c set" shows "Always A = {F. reachable F \<subseteq> A}"
triangle_right: shows "Cop.H \<epsilon>' g \<cdot> \<a>[g, f', g] \<cdot> Cop.H g \<eta>' = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
assms_1: shows "z \<noteq> \<infinity>\<^sub>h"
Cod_Inv: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t" shows "bicategorical_language.Cod V src trg (bicategorical_language.Inv V t) = bicategorical_language.Dom V src trg t"
dvd_imp_degree_le: fixes p :: "'c poly"   and q :: "'c poly" assumes "p dvd q"   and "q \<noteq> 0" shows "degree p \<le> degree q"
Preflow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "Preflow c s t f \<equiv> (\<forall>e. (0::'a) \<le> f e \<and> f e \<le> c e) \<and> (\<forall>v\<in>Graph.V c - {s, t}. sum f (Graph.outgoing c v) \<le> sum f (Graph.incoming c v))"
brec_iqm_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "brec_iqm \<delta> \<equiv> ls.iterate \<delta> (\<lambda>r m. if rhsq r = [] then hm.update (lhs r) (NODE (rhsl r) []) m else m) (hm.empty ())"
wlsFSb_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) Recursion.model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.wlsFSb Delta MOD \<equiv> FixSyn.gWlsAllDisj Delta MOD \<and> FixSyn.gWlsAbsIsInBar Delta MOD \<and> FixSyn.gConsPresGWls Delta MOD \<and> FixSyn.gSubstAllPresGWlsAll Delta MOD \<and> FixSyn.gFreshCls Delta MOD \<and> FixSyn.gSubstCls Delta MOD \<and> FixSyn.gAbsRen Delta MOD"
group_l_invI: assumes "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<oplus> x = \<zero>" shows "Group.group (add_monoid G)"
carrier_vecD: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "dim_vec v = n"
share_all_until_volatile_write_Ghost\<^sub>s\<^sub>b_commute: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, Ghost\<^sub>s\<^sub>b A L R W # sb, \<D>, \<O>, \<R>)"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)) \<inter> (R \<union> L \<union> A) = {}" shows "share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb, \<D>', \<O>', \<R>')]) (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) = share_all_until_volatile_write ts \<S>"
is_unit_mult_iff: fixes a :: "'a"   and b :: "'a" shows "(a * b dvd (1::'a)) = (a dvd (1::'a) \<and> b dvd (1::'a))"
triangle_in_hom_2: shows "\<guillemotleft>(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g) : trg\<^sub>D (F g) \<star>\<^sub>D F g \<Rightarrow>\<^sub>D F g \<star>\<^sub>D src\<^sub>D (F g)\<guillemotright>"
cut_less_Un: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<union> B) \<down>< t = A \<down>< t \<union> B \<down>< t"
assms_2: assumes "\<And>\<eta>. exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g \<A> \<eta> \<surd>" shows "Negligible.negligible adv"
m_concat_append: fixes a :: "'a list"   and b :: "'a list" assumes "set a \<subseteq> carrier G"   and "set b \<subseteq> carrier G" shows "m_concat (a @ b) = m_concat a \<otimes> m_concat b"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" shows "redT_updTs' ts [] = ts"
vcons_def: fixes xs :: "V"   and x :: "V" shows "xs #\<^sub>\<circ> x = vinsert \<langle>vcard xs, x\<rangle> xs"
CConcEnd1: fixes q :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "q @ p \<noteq> []"   and "x \<notin> dom (Cp a)" shows "Cp (list2FWpolicy (q @ p @ [a])) x = Cp (list2FWpolicy (q @ p)) x"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_unfold: fixes A :: "'a set" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> SKIP |-| STOP |-| Mprefix A (\<lambda>x. CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A)"
count_inI: fixes M :: "'c multiset"   and x :: "'c" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
bezout_matrix_iarrays_def: fixes A :: "'b iarray iarray"   and a :: "nat"   and b :: "nat"   and j :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'c" shows "bezout_matrix_iarrays A a b j bezout = tabulate2 (nrows_iarray A) (nrows_iarray A) (let (p, q, u, v, d) = bezout (A !! a !! j) (A !! b !! j) in (\<lambda>x y. if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then 1::'a else (0::'a)))"
action_proj_def: fixes a :: "('b, 'c) fmap \<times> ('b, 'd) fmap"   and vs :: "'b set" shows "action_proj a vs \<equiv> (fmrestrict_set vs (fst a), fmrestrict_set vs (snd a))"
Refl_Restr: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Refl r" shows "Refl (Restr r A)"
stable_onI: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.individually_rational_on Pd Ch ds X"   and "Contracts.stable_no_blocking_on Pd Ch ds X" shows "Contracts.stable_on Pd Ch ds X"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and T1 :: "'c trm"   and T2 :: "'c trm" shows "substA \<pi> (Eq T1 T2) = Eq (subst \<pi> T1) (subst \<pi> T2)"
odd_imp_coprime_nat: fixes n :: "nat" assumes "odd n" shows "coprime n 2"
elims: fixes x :: "'fsym trm"   and y :: "('fsym, 'tp) efsym trm" assumes "tNN x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y = (if unprot (TE.tpOfV xa) \<or> protFw (TE.tpOfV xa) then Var xa else Fn (Tag (TE.tpOfV xa)) [Var xa])"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "y = (if unprot (resOf f) \<or> protFw (resOf f) then Fn (Oldf f) (map tNN Tl) else Fn (Tag (resOf f)) [Fn (Oldf f) (map tNN Tl)])"
fps_X_mult_right_nth: fixes a :: "'b fps"   and n :: "nat" shows "(a * fps_X) $ n = (if n = 0 then 0::'b else a $ (n - 1))"
i_set_i_set_ind_eq: shows "i_set = i_set_ind"
set_awalk_verts_sd_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and e :: "'a \<times> 'a"   and w :: "'a" assumes "S.apath u p v"   and "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)" shows "set (pawalk_verts u (sd_path e w p)) \<subseteq> set (pawalk_verts u p) \<union> {w}"
push_down_state_rank_tokens: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some i" shows "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {}"
extract_var_finite_set: fixes p :: "'b mpoly"   and S :: "(nat \<Rightarrow>\<^sub>0 nat) set"   and v :: "nat" assumes "{m'. coeff p m' \<noteq> (0::'b)} \<subseteq> S"   and "finite S" shows "extract_var p v = (\<Sum>m\<in>S. monom (remove_key v m) (monom (Poly_Mapping.single v (lookup m v)) (coeff p m)))"
simps_5: fixes f :: "Core.term"   and u :: "Core.term"   and f' :: "Core.term"   and u' :: "Core.term"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" shows "assoc_match_term (f $ u) (f' $ u') subs = Option.bind (assoc_match_term f f' subs) (assoc_match_term u u')"
bounded_minus_comp: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x - g x) ` S)"
S2: shows "(\<exists>n1 x1. c_pair n x = c_pair n1 x1 \<and> x1 \<in> nat_to_ce_set n1) = (x \<in> nat_to_ce_set n)"
continuous_at_imp_continuous_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>s. isCont f x" shows "continuous_on s f"
ints_2: shows "integrable M (\<lambda>w. max 0 (- X w))"
accp_acc_eq: fixes r :: "('b \<times> 'b) set" shows "Wellfounded.accp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Wellfounded.acc r)"
WellOrderedStrategies_axioms: shows "WellOrderedStrategies G S p good r"
ndef: shows "n = nat \<lceil>2 * (\<Sum>n. 1 / real (Suc n))\<rceil>"
vector_space_ow_def: fixes U\<^sub>V\<^sub>S :: "'d set"   and plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and minus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<equiv> ab_group_add_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S \<and> vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale"
icard_finite_conv: fixes A :: "'a set" shows "(icard A = enat (card A)) = finite A"
induct: fixes P :: "nat \<Rightarrow> (nat \<times> nat) list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "(nat \<times> nat) list"   and a2 :: "nat" assumes "\<And>u v. P u [] v"   and "\<And>u x y p v. P y p v \<Longrightarrow> P u ((x, y) # p) v" shows "P a0 a1 a2"
orecvmsg_simps_8: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd" shows "orecvmsg P \<sigma> \<tau>\<^sub>s = True"
scale_minus_both: fixes a :: "'a"   and x :: "'a" shows "- a * - x = a * x"
diamond_def: shows "diamond \<equiv> \<lambda>\<phi>. \<^bold>\<not>\<^bold>\<box>\<^bold>\<not>\<phi>"
functor_axioms: shows "functor J (\<cdot>\<^sub>A) D"
of_int: fixes x :: "int" assumes "0 \<le> x"   and "x < p" shows "R (arith_ops_record.of_int ff_ops x) (of_int x)"
assoc_mult_mat_vec: fixes A :: "'b mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'b mat"   and n\<^sub>3 :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "v \<in> carrier_vec n\<^sub>3" shows "A * B *\<^sub>v v = A *\<^sub>v (B *\<^sub>v v)"
new_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'aj" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
spmf_map_restrict: fixes p :: "('b \<times> 'c) spmf"   and y :: "'c"   and x :: "'b" shows "spmf (map_spmf fst (p \<upharpoonleft> (snd -` {y}))) x = spmf p (x, y)"
statClosed: fixes \<Psi> :: "'b"   and \<phi> :: "'c"   and p :: "(name \<times> name) list" assumes "\<Psi> \<turnstile> \<phi>" shows "p \<bullet> \<Psi> \<turnstile> p \<bullet> \<phi>"
map_def: fixes f :: "'b" shows "B.map f = (if B.arr f then f else B.null)"
F_D_part: fixes P :: "'a process" shows "F P = {(s, x). s \<in> D P} \<union> {(s, x). s \<notin> D P \<and> (s, x) \<in> F P}"
hyp: fixes A :: "(('a, 'n) vec, 'm) vec" assumes "n < nrows A" shows "\<exists>P. invertible P \<and> row_add_iterate A n i j = P ** A"
nat_ivl_Suc': fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> Suc n" shows "prod g {m..Suc n} = g (Suc n) * prod g {m..n}"
assms: shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
subgroup_self: shows "subgroup (carrier (alt_group n)) (alt_group n)"
cycle_free_diag_intro: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'a) \<le> len m i i xs" shows "cycle_free m n"