hcju/isabelleps huggingface dataset:
case_one_psi_total: fixes j :: "nat"   and x :: "nat" assumes "\<nexists>t. local.prefixes t j \<up>"   and "0 < x" shows "\<psi> j x \<down>= 0 \<or> \<psi> j x \<down>= 1"
inv_def: fixes f :: "'d \<Rightarrow> 'c" shows "inv f = (\<lambda>y. SOME x. f x = y)"
induct: fixes base :: "nat"   and x :: "hmultiset"   and P :: "hmultiset \<Rightarrow> bool" assumes "2 \<le> base"   and "well_base\<^sub>h base x"   and "\<And>M. \<lbrakk>\<forall>N\<in>#hmsetmset M. well_base\<^sub>h base N \<and> P N; well_base base (hmsetmset M)\<rbrakk> \<Longrightarrow> P M" shows "P x"
commute: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "sup_prod a b = sup_prod b a"
Union_rel_br: fixes l :: "'c \<Rightarrow> 'e set"   and lI :: "'c \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'd set"   and sI :: "'e \<Rightarrow> bool" shows "\<langle>br l lI, br s sI\<rangle>Union_rel = br (\<lambda>a. \<Union> (s ` l a)) (\<lambda>a. lI a \<and> (\<forall>c\<in>l a. sI c))"
dims1_def: shows "st2.dims1 = nths dims {Suc k}"
ord_eq_le_eq_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a = b"   and "b \<le> c"   and "c = d" shows "a \<le> d"
closed_levelset_within: fixes S :: "'c set"   and s :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "continuous_on S s"   and "closed S" shows "closed {x \<in> S. s x = c}"
po_antisym: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "porder le"   and "le x y"   and "le y x" shows "x = y"
assms: fixes s :: "'a" shows "\<I> \<turnstile>c oracle s \<surd>"
f_bounded: fixes t :: "real"   and x :: "'a" assumes "t \<in> T"   and "x \<in> X'" shows "norm (f t x) \<le> B"
IH_7: assumes "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set A" shows "\<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) \<or>\<notin>: G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
list_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>list_rel \<equiv> {(l, l'). list_all2 (\<lambda>x x'. (x, x') \<in> R) l l'}"
ennreal_divide_eq_top_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b = \<top>) = (a \<noteq> 0 \<and> b = 0 \<or> a = \<top> \<and> b \<noteq> \<top>)"
subgroup_kernel: shows "subgroup (kernel (G\<lparr>carrier := generate G {a}\<rparr>) (H\<lparr>carrier := generate H {b}\<rparr>) x) (G\<lparr>carrier := generate G {a}\<rparr>)"
assms_1: shows "spec_ops (xs @ [(oid, oper)]) deps"
simps: fixes i :: "'a"   and j :: "nat" shows "received_messages (i, j) = concat (map (case_event (\<lambda>message. []) (\<lambda>x m. [m])) (take j (events i)))"
walk_first_edge': fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list" assumes "Digraph G"   and "Digraph.walk G (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow>\<^bsub>G\<^esub> hd xs"
2: shows "\<guillemotleft>unit (trg\<^sub>C f) \<cdot>\<^sub>D \<epsilon>' \<cdot>\<^sub>D D.inv (\<Phi> (f, g)) : F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
alt_sem_lemma: fixes ODE :: "('d, 'e) ODE"   and I :: "('d, 'f, 'e) interp"   and sol :: "real \<Rightarrow> (real, 'e) vec"   and t :: "real"   and ab :: "(real, 'e) vec" assumes "osafe ODE" shows "ODE_sem I ODE (sol t) = ODE_sem I ODE (\<chi>i. if i \<in> FVO ODE then sol t $ i else ab $ i)"
abs_summable_on_norm_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "((\<lambda>x. norm (f x)) abs_summable_on A) = (f abs_summable_on A)"
Direct: fixes new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl"   and G :: "prog"   and superNew :: "qtname" assumes "\<not> is_static new"   and "msig new = msig old"   and "G\<turnstile>Method new declared_in declclass new"   and "G\<turnstile>Method old declared_in declclass old"   and "G \<turnstile>Method old inheritable_in pid (declclass new) "   and "G\<turnstile>declclass new\<prec>\<^sub>C1superNew"   and "G \<turnstile>Method old member_of superNew" shows "G \<turnstile> new overrides\<^sub>S old"
plus_\<I>_sel_1: fixes \<I>l :: "('a, 'c) \<I>"   and \<I>r :: "('b, 'd) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
assms_2: shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
Ch_unilateral_substitutes: shows "\<forall>h. ManiPref'.unilateral_substitutes_on UNIV (Ch h)"
T_F_spec: fixes t :: "'a event list"   and P :: "'a process" shows "((t, {}) \<in> F P) = (t \<in> T P)"
allE: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x. P x" obtains "P x"
errMOD_igWlsAbsDisj: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igWlsAbsDisj MOD" shows "igWlsAbsDisj (errMOD MOD)"
bind_bind: fixes A :: "'f set"   and B :: "'f \<Rightarrow> 'e set"   and C :: "'e \<Rightarrow> 'd set" shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
card_V_eq_a: fixes V :: "nat set"   and a :: "nat" assumes "V \<subseteq> {0..<a}" shows "(card V = a) = (V = {0..<a})"
init: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e" assumes "(\<sigma>, p) \<in> init A"   and "l \<in> labels \<Gamma> p" shows "P (\<sigma>, l)"
aux_3: fixes t :: "nat"   and X :: "event_data option list set" assumes "(t, X) \<in> set auxlist" shows "ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne - 1) (Sincep pos \<phi> (point (\<tau> \<sigma> (ne - 1) - t)) \<psi>)) X"
max_list_Cons: fixes x :: "'b"   and xs :: "'b list" shows "Missing_Lemmas.max_list (x # xs) = (if xs = [] then x else max x (Missing_Lemmas.max_list xs))"
select_convs_6: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "state_t.more \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = more"
q'_4: fixes K :: "'n set" assumes "K \<in> q" shows "\<exists>a b. K = cbox a b"
divide_less_eq_numeral_1: fixes b :: "'a"   and c :: "'a"   and w :: "num" shows "(b / c < numeral w) = (if (0::'a) < c then b < numeral w * c else if c < (0::'a) then numeral w * c < b else (0::'a) < numeral w)"
simps_2: fixes x :: "char list"   and a :: "aexp"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT (x ::= a) e = (\<lambda>s. e (s[a/x]))"
I_in_Gen_P: shows "I \<subseteq> Gen (\<P> k u)"
empty: fixes t :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" shows "trclp t c [] c"
minus_add: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = \<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> y"
zip_map_fst_snd: fixes zs :: "('g \<times> 'h) list" shows "zip (map fst zs) (map snd zs) = zs"
wffp_cases2: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" obtains n :: "nat" and st :: "'St" where "0 < na__" and "sl = map Idle (replicate na__ sta__)"   | n :: "nat" and st :: "'St" and st1 :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "sl = map Idle (replicate n st) @ [State st1 u c] @ sl1" and "sl1 \<noteq> [] \<Longrightarrow> wffp sl1 \<and> hd sl1 \<in> \<delta> (State st1 u c)"
select_convs_1: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.states \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = states"
cbig_big_equiv: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state" shows "(\<exists>coll. (\<sigma>', coll) \<in> cbig P \<sigma>) = (\<sigma>' \<in> big P \<sigma>)"
hs_get_work_locs_def: shows "hs_get_work_locs = prefixed ''mark_loop_get_work'' - {mark_loop_get_work_init_type}"
coeff_degree_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "lead_coeff (p * q) = lead_coeff q * lead_coeff p"
coreProperties: fixes V :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set"   and I :: "nat set"   and J :: "nat set" assumes "2 \<le> card V"   and "getIJ Q_pos Q_neg V = (I, J)" shows "1 \<le> card I \<and> 1 \<le> card J \<and> V = I \<union> J \<and> I \<inter> J = {}"
split_list_first: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
sets_vimage_Sup_eq: fixes M :: "'b measure set"   and f :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and Y :: "'b set" assumes "M \<noteq> {}"   and "f \<in> X \<rightarrow> Y"   and "\<And>m. m \<in> M \<Longrightarrow> space m = Y" shows "sets (vimage_algebra X f (Sup M)) = sets (Sup (vimage_algebra X f ` M))"
inj_semigroup_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_semigroup_add_hom hom \<equiv> semigroup_add_hom hom \<and> injective hom"
blinfun_apply_Pair_split: fixes g :: "('c \<times> 'd) \<Rightarrow>\<^sub>L 'b"   and a :: "'c"   and b :: "'d" shows "blinfun_apply g (a, b) = blinfun_apply g (a, 0::'d) + blinfun_apply g (0::'c, b)"
cINF_lower: fixes f :: "'b \<Rightarrow> 'a multiset"   and A :: "'b set"   and x :: "'b" assumes "subset_mset.bdd_below (f ` A)"   and "x \<in> A" shows "\<Sqinter> (f ` A) \<subseteq># f x"
SDC: shows "(([u], c), LCall p, [entry fg p, u'], c) \<in> trss fg"
exhaust: fixes y :: "X5" obtains "y = X5.Xd1"   | "y = X5.Xd1'"   | "y = X5.Xd2"   | "y = X5.Xd2'"   | "y = Xd2''"
cases: fixes r :: "('b, 'c) AlgState" obtains aActs :: "'b" and aTrans :: "'c" where "r = \<lparr>aActs = aActs, aTrans = aTrans\<rparr>"
RDirProd_add_monoid: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "add_monoid (RDirProd R S) = add_monoid R \<times>\<times> add_monoid S"
right_total_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "right_total R = ((=) \<le> R\<inverse>\<inverse> OO R)"
neville_aitken_main: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'b"   and i :: "nat"   and k :: "nat"   and j :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>i j. \<lbrakk>i < j; j \<le> n\<rbrakk> \<Longrightarrow> x i \<noteq> x j"   and "i \<le> k"   and "k \<le> j"   and "j \<le> n" shows "poly (neville_aitken_main x f i j) (x k) = f k"
ctr: fixes a :: "'b" shows "Generative_Probabilistic_Value.Done a = GPV (return_spmf (Pure a))"
lexordp_eq_refl: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "ord.lexordp_eq less xs xs"
count_union: fixes M :: "'d multiset"   and N :: "'d multiset"   and a :: "'d" shows "count (M + N) a = count M a + count N a"
tObsC_tLength: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" assumes "tObsC t = tObsC t'" shows "tLength t = tLength t'"
cases: fixes a :: "'val"   and g :: "'g"   and \<phi> :: "'val" assumes "a \<in> reachable g \<phi>" obtains "a = \<phi>" and "unnecessaryPhi g \<phi>"   | \<phi>'' :: "'val" and \<phi>''' :: "'val" where "a = \<phi>'''" and "\<phi>'' \<in> reachable g \<phi>" and "phiArg g \<phi>'' \<phi>'''" and "unnecessaryPhi g \<phi>'''"
lprefixes_chain: fixes xs :: "'c llist" shows "Complete_Partial_Order.chain (\<sqsubseteq>) {ys. ys \<sqsubseteq> xs}"
iT_Mult_image_conv: fixes I :: "nat set"   and k :: "nat" shows "I \<otimes> k = (\<lambda>n. n * k) ` I"
simps_1: fixes r1 :: "'b rexp"   and r2 :: "'b rexp" shows "simp (Plus r1 r2) = simp_Plus (simp r1) (simp r2)"
mono_neutral_cong_right: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'b)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g T = prod h S"
ord_rec_non: fixes x :: "hf"   and T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf" assumes "\<not> Ord x" shows "ord_rec T G H x = H x"
pth_7_1: fixes x :: "'a" shows "(0::'a) + x \<equiv> x"
simps_8: fixes d :: "'a \<Rightarrow> ('a, 'b, 'c) com" shows "redex (DynCom d) = DynCom d"
push_down_state_rank_token_run: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and i :: "nat" assumes "\<MM>.state_rank q n = Some i" shows "\<exists>x. \<MM>.token_run x n = q \<and> x \<le> n"
interchange_columns_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and m :: "'e"   and i :: "'f" shows "interchange_columns_row A n m i = (\<chi>j. if j = n then A $ i $ m else if j = m then A $ i $ n else A $ i $ j)"
cong_mult_rcancel: fixes k :: "'a"   and m :: "'a"   and a :: "'a"   and b :: "'a" assumes "coprime k m" shows "[a * k = b * k] (mod m) = [a = b] (mod m)"
lossless_gpvI: fixes gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "lossless_spmf (the_gpv gpv)"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> lossless_gpv \<I> (c input)" shows "lossless_gpv \<I> gpv"
"1_2": fixes ass :: "'a \<Rightarrow> bool" shows "val_ifex i ass = False"
ZObis_raw_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> ZOretr theta" shows "theta \<subseteq> ZObis"
arg_cong2: fixes a :: "'f"   and b :: "'f"   and c :: "'g"   and d :: "'g"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" assumes "a = b"   and "c = d" shows "f a c = f b d"
primorial'_def: fixes n :: "nat" shows "primorial' n = prod nth_prime {..<n}"
bind_lifted_simps_2: fixes x :: "'b"   and f :: "'b \<rightarrow> 'a\<cdot>lifted" shows "Lifted\<cdot>x \<bind> f = f\<cdot>x"
eq: fixes n :: "nat" shows "outer_measure_of M (A n) = outer_measure_of M (F n)"
cases: fixes a :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "a \<in> \<alpha>.active_threads s" obtains t :: "'t" and x :: "'x" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x'm' :: "'x \<times> 'm" and ln :: "'l \<Rightarrow>f nat" where "a = ta__" and "thr s ta__ = \<lfloor>(x__, ln__)\<rfloor>" and "ln__ = no_wait_locks" and "pred.eval (r ta__ (x__, shr s)) (ta, x'm')" and "\<alpha>.actions_ok s ta__ ta"   | t :: "'t" and x :: "'x" and ln :: "'l \<Rightarrow>f nat" where "a = t" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "ln \<noteq> no_wait_locks" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln"
wffp_append_iff: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list"   and sl1 :: "('a, 'b, 'c) state list" shows "GM_sec_model.wffp do (sl @ sl1) = (GM_sec_model.wffp do sl \<and> sl1 = [] \<or> sl = [] \<and> GM_sec_model.wffp do sl1 \<or> GM_sec_model.wffp do sl \<and> GM_sec_model.wffp do sl1 \<and> hd sl1 \<in> GM_sec_model.\<delta> do (last sl))"
iter_narrow_pfp: fixes f :: "'a acom \<Rightarrow> 'a acom"   and c0 :: "'a acom"   and c :: "'a acom" assumes "preord_class.mono f"   and "f c0 \<sqsubseteq> c0"   and "iter_narrow f c0 = Some c" shows "f c \<sqsubseteq> c \<and> c \<sqsubseteq> c0"
simps_2: fixes k :: "nat" shows "HTuple (Suc k) = HPair Zero (HTuple k)"
linear_span_iOp_closedTr2: fixes A :: "'b set"   and H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "s \<in> {j. j \<le> n} \<rightarrow> A" shows "-\<^sub>a l_comb R M n s f = l_comb R M n (\<lambda>x\<in>{j. j \<le> n}. -\<^sub>a\<^bsub>R\<^esub> s x) f"
cINF_greatest: fixes A :: "'a set"   and m :: "'b multiset"   and f :: "'a \<Rightarrow> 'b multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># Inf (f ` A)"
Arr_iff_cod_in_codomain: fixes f :: "'a" shows "right a f = (cod f \<in> R.codomains f)"
induct: fixes x1 :: "Expr"   and x2 :: "TP"   and P :: "Expr \<Rightarrow> TP \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_expr"   and "\<And>x t. CONTEXT x = t \<Longrightarrow> P (varE x) t"   and "\<And>v. v = RVal Nullref \<or> (\<exists>i. v = IVal i) \<Longrightarrow> P (valE v) low"   and "\<And>e1 t e2 f. \<lbrakk>(e1, t) \<in> VS_expr; P e1 t; (e2, t) \<in> VS_expr; P e2 t; opEGood f\<rbrakk> \<Longrightarrow> P (opE f e1 e2) t"   and "\<And>e. P e high" shows "P x1 x2"
dh: fixes x :: "complex" assumes "x \<in> S" shows "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within S))"
set_to_map_empty_iff_2: fixes S :: "('g \<times> 'h) set" shows "(Map.empty = set_to_map S) = (S = {})"
diagonal_step_PQ_def: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_step_PQ A i k bezout = (let i_row = from_nat i; k_row = from_nat k; i_col = from_nat i; k_col = from_nat k; (p, q, u, v, d) = bezout (A $h i_row $h from_nat i) (A $h k_row $h from_nat k); P = row_add (interchange_rows (row_add (mat (1::'b)) k_row i_row p) i_row k_row) k_row i_row (- v) in Let (mult_column (column_add (column_add (mat (1::'b)) i_col k_col q) k_col i_col u) k_col (- (1::'b))) (Pair P))"
in_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d list"   and b :: "'e list" shows "list_all2 R a b = (\<exists>z. z \<in> {x. set x \<subseteq> {(x, y). R x y}} \<and> map fst z = a \<and> map snd z = b)"
e_def: shows "e = r - of_int \<lfloor>r\<rfloor>"
union_ring_carrier: fixes C :: "('c, 'd) ring_scheme set" shows "carrier (union_ring C) = \<Union> (carrier ` C)"
StaticUpCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s'\<rangle>"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Ds),s'\<rangle>"
uminus_eq_mat: fixes A :: "'b mat"   and B :: "'b mat" shows "(- A = - B) = (A = B)"
full_pmdlD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "'k set"   and v :: "'t" assumes "p \<in> full_pmdl K"   and "v \<in> keys p" shows "component_of_term v \<in> K"
zip_eq: fixes u :: "'b list"   and v :: "'c list"   and r :: "'b list"   and s :: "'c list" assumes "length u = length v"   and "length r = length s" shows "(u || v = r || s) = (u = r \<and> v = s)"
ordLeq_refl: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "r \<le>o r"
pdevs_val_sum_le: fixes X :: "'a pdevs"   and d :: "nat"   and e :: "nat \<Rightarrow> real" assumes "degree X \<le> d" shows "pdevs_val e X = (\<Sum>i<d. e i *\<^sub>R pdevs_apply X i)"
li_le_dim_1: fixes A :: "'a vec set" assumes "fin_dim"   and "A \<subseteq> carrier_vec n"   and "lin_indpt A" shows "finite A"
bi_uniqueI: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique R"   and "right_unique R" shows "bi_unique R"
Quotient_st_mat: shows "Quotient (eq_onp (\<lambda>a. non_neg_mat a \<and> right_stoch_mat a)) Abs_st_mat st_mat cr_st_mat"
det_upper_triangular: fixes A :: "'a mat"   and n :: "nat" assumes "Matrix.upper_triangular A"   and "A \<in> carrier_mat n n" shows "Determinant.det A = prod_list (diag_mat A)"
dfs_invariant: fixes S :: "('ma, 'mt) AlgState"   and xs :: "'rep list" assumes "k_invariant S"   and "list_all k_isNode xs" shows "k_invariant (KBPAlg.dfs S xs)"
omega_lfp: 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 one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and post :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.complete_mbt_algebra Inf Sup inf less_eq less sup bot top one times dual dual_star omega star post" shows "times (omega x) y = complete_lattice.lfp Inf less_eq (\<lambda>z. inf (times x z) y)"
induct: fixes P :: "(int \<Rightarrow> 'b) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int \<Rightarrow> 'b"   and a1 :: "int"   and a2 :: "int" assumes "\<And>a l h. (l < h \<Longrightarrow> P a (l + 1) h) \<Longrightarrow> P a l h" shows "P a0 a1 a2"
A2: shows "A2 \<in> carrier_mat 1 (n - 1)"
outstanding_refs_non_empty_dropWhile: fixes P :: "'b memref \<Rightarrow> bool"   and xs :: "'b memref list" assumes "outstanding_refs P xs \<noteq> {}" shows "outstanding_refs P (dropWhile (Not \<circ> P) xs) \<noteq> {}"
fmrestrict_set_dom: fixes m :: "('a, 'b) fmap" shows "fmrestrict_set (fmdom' m) m = m"
assms_1: shows "us \<in> lists \<langle>G\<rangle>\<^sub>+"
continuous_map_id: fixes X :: "'d topology" shows "continuous_map X X id"
perm_simps_3: fixes p :: "perm"   and var :: "Syntax.var"   and "term" :: "Syntax.term" shows "p \<bullet> Lam var term = Lam (p \<bullet> var) (p \<bullet> term)"
symvars_def: fixes s :: "'c \<Rightarrow> nat" shows "symvars s = range (\<lambda>v. symvar v s)"
star_class_defs_4: shows "(-) \<equiv> *f2* (-)"
failingSubSubs: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b" assumes "inherited subs P"   and "\<not> P (tree subs gamma)"   and "\<not> terminal subs gamma"   and "fans subs" shows "failingSub subs P gamma \<in> subs gamma"
isomorphic_reduced_homology_by_contractible: fixes X :: "'b topology"   and S :: "'b set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "reduced_homology_group p X \<cong> relative_homology_group p X S"
g: shows "(a, b) \<in> set (shift (adjust (tm_of abc_fourtimes @ shift (mopup (Suc 0)) (length (tm_of abc_fourtimes) div 2)) (Suc ((length (tm_of abc_fourtimes) + 16) div 2))) (length t_twice div 2 + 13))"
mem_rev_existence_ivl_eq: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" shows "(t \<in> ll_on_open.existence_ivl (preflect t0 ` T) (\<lambda>t. - f (preflect t0 t)) X t0 x0) = (preflect t0 t \<in> existence_ivl t0 x0)"
backlinks_alt: fixes E :: "('b \<times> 'b) list" shows "backlinks E = map (\<lambda>e. (snd e, fst e)) E"
map_fst_enumerate: fixes n :: "nat"   and xs :: "'g list" shows "map fst (enumerate n xs) = [n..<n + length xs]"
matrix_add_0_left: fixes A :: "('c, 'd, 'e) matrix" shows "0 + A = A"
N_2: fixes n :: "nat" assumes "N \<le> n" shows "Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. C0 \<le> \<bar>x\<bar>} < epsilon"
Sup_nat_def: fixes X :: "nat set" shows "Sup X = (if X = {} then 0 else Max X)"