hcju/isabelleps huggingface dataset:
prod_n_ideal_cont2: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and P :: "'a set" assumes "\<forall>l\<le>n. ideal R (J l)"   and "prime_ideal R P"   and "\<Inter> {X. \<exists>k\<le>n. X = J k} \<subseteq> P" shows "\<exists>l\<le>n. J l \<subseteq> P"
ssubst_vquot_Ex: fixes V :: "name set"   and i :: "name"   and A :: "fm" assumes "finite V"   and "atom i \<sharp> p' \<bullet> V" shows "ssubst \<lfloor>SyntaxN.Ex i A\<rfloor>(insert i V) (insert i V) F' = ssubst \<lfloor>SyntaxN.Ex i A\<rfloor>V V F'"
sq_norm_of_int: fixes v :: "int vec" shows "\<parallel>of_int_vec v\<parallel>\<^sup>2 = of_int \<parallel>v\<parallel>\<^sup>2"
is_non_volatile_Write\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_non_volatile_Write\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = (\<not> volatile)"
hinter_hempty_right: fixes A :: "hf" shows "A \<sqinter> 0 = 0"
Cauchy_convergent_iff: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = convergent X"
triangle_nextVertex_prevVertex: fixes f :: "face"   and a :: "nat" assumes "triangle f"   and "a \<in> \<V> f"   and "distinct (vertices f)" shows "f \<bullet> (f \<bullet> a) = f\<^bsup>-1\<^esup> \<bullet> a"
mupdate_def: fixes v :: "'V"   and wv :: "'W infty"   and \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)" shows "mupdate v wv \<sigma> \<equiv> ASSERT (update_pre v (\<alpha>s \<sigma>) \<and> wv = mpath_weight' (snd \<sigma> v)) \<bind> (\<lambda>_. let (wl, res) = \<sigma>; pv = mpath' (res v) in FOREACH\<^bsup>uinvarm v (\<alpha>w wl) (\<alpha>r res)\<^esup> (succ G v) (\<lambda>(w', v') (wl, res). if wv + Num w' < mpath_weight' (res v') then ASSERT (v' \<in> dom wl \<and> pv \<noteq> None) \<bind> (\<lambda>_. ASSERT (wv \<noteq> infty.Infty) \<bind> (\<lambda>_. RETURN (wl(v' \<mapsto> wv + Num w'), res(v' \<mapsto> ((v, w', v') # the pv, val wv + w'))))) else RETURN (wl, res)) (wl, res))"
lex_comp': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "lex_comp' x y = comp_of_ord lex_pp (rep_nat_pp x) (rep_nat_pp y)"
count_notin: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "count_list xs x = 0"
d: shows "R1 \<subseteq> upRules \<and> R = Ax \<union> R1"
simps_1: fixes n :: "nat"   and c :: "poly" shows "tmbound n (CP c) = True"
lunit_simps_4: fixes f :: "'a" assumes "B.ide f" shows "B.dom (B.lunit f) = trg\<^sub>B f \<star>\<^sub>B f"
zero: fixes n :: "nat" shows "ind (is_nth_power n) 0 = (0::'b)"
induct: fixes x1 :: "('t + 'n) list fset"   and x2 :: "'t list"   and P :: "('t + 'n) list fset \<Rightarrow> 't list \<Rightarrow> bool" assumes "local.in_cfls x1 x2"   and "\<And>P. fBex P local.\<oo>\<^sub>P \<Longrightarrow> P P []"   and "\<And>a P w. \<lbrakk>local.in_cfls (ffUnion ((\<lambda>\<alpha>. local.\<dd>\<^sub>P \<alpha> a) |`| P)) w; P (ffUnion ((\<lambda>\<alpha>. local.\<dd>\<^sub>P \<alpha> a) |`| P)) w\<rbrakk> \<Longrightarrow> P P (a # w)" shows "P x1 x2"
no_zero_divisors: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors plus zero times"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "times a b \<noteq> zero"
subset_iff_psubset_eq: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq B A = (order.greater B A \<or> A = B)"
theI_unique: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<exists>!x. P x" shows "P x = (x = The P)"
Bij: fixes x :: "nat"   and y :: "nat" assumes "x < n"   and "y < n"   and "x \<noteq> y" shows "B $$ (x, y) = (0::'a)"
lub_insert: fixes x :: "'e"   and S :: "'e set" shows "lub (insert x S) = x \<squnion> lub S"
IQ: shows "SkewBinomialHeapStruc.invar (map bsmapt q)"
null_char: shows "Fop.Bop.null = Cop.null"
simps_1: fixes uu :: "'b" shows "del uu N0 = N0"
p_max: fixes x :: "real" shows "p x \<le> p_max"
distr_lift_RealVal: fixes f :: "real \<Rightarrow> real"   and M :: "val measure"   and \<delta> :: "val \<Rightarrow> ennreal"   and g :: "(real \<Rightarrow> ennreal) \<Rightarrow> real \<Rightarrow> ennreal" assumes "f \<in> borel_measurable borel"   and "has_subprob_density M (stock_measure REAL) \<delta>"   and "\<And>M \<delta>. has_subprob_density M lborel \<delta> \<Longrightarrow> has_density (distr M borel f) lborel (g \<delta>)" shows "has_density (distr M (stock_measure REAL) (lift_RealVal f)) (stock_measure REAL) (g (\<lambda>x. \<delta> (RealVal x)) \<circ> extract_real)"
subformulas\<^sub>\<nu>_semantics: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<nu> \<phi> = {\<psi> \<in> subfrmlsn \<phi>. \<exists>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi> = \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<or> \<psi> = \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2}"
snapshot_stable: fixes t :: "('a, 'b, 'c) event llist"   and i :: "nat"   and j :: "nat"   and p :: "'a" assumes "cotrace init t"   and "i \<le> j"   and "ps (coS t i) p \<noteq> None" shows "ps (coS t j) p \<noteq> None"
Re_Arccos_bounds: fixes z :: "complex" shows "- pi < Re (Arccos z) \<and> Re (Arccos z) \<le> pi"
open_neighbourhood_base_of: fixes P :: "'c set \<Rightarrow> bool"   and X :: "'c topology" assumes "\<forall>S. P S \<longrightarrow> openin X S" shows "neighbourhood_base_of P X = (\<forall>W x. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U. P U \<and> x \<in> U \<and> U \<subseteq> W))"
comp_char: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "COMP g f = (if seq g f then MkArr (Dom f) (Cod g) (Comp (Cod g) (Dom g) (Dom f) (Map g) (Map f)) else null)"
resolve_rename_either_way_def: fixes C :: "'a literal list"   and D :: "'a literal list" shows "resolve_rename_either_way C D = resolve_rename C D @ resolve_rename D C"
uminus_sum_list_map: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (monoid_add.sum_list plus zero (map f xs)) = monoid_add.sum_list plus zero (map (uminus \<circ> f) xs)"
simps_2: fixes x :: "tyvrs"   and y :: "ty" shows "tyvrs_of (TVarB x y) = {x}"
simps: fixes uu :: "'a extB" shows "uu\<^sup>\<star> = extB.Val top"
complete\<^sub>N_I': fixes c :: "nat \<Rightarrow> real"   and N :: "'b quasinorm" assumes "\<And>n. 0 < c n"   and "\<And>u. \<lbrakk>\<forall>n. u n \<in> space\<^sub>N N; \<forall>n. Norm N (u n) \<le> c n\<rbrakk> \<Longrightarrow> \<exists>x\<in>space\<^sub>N N. tendsto_in\<^sub>N N (\<lambda>n. sum u {0..<n}) x" shows "complete\<^sub>N N"
assms_2: shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>"
\<theta>_simps_3: shows "trg \<theta>' = src (k\<^sup>* \<star> h)"
bijective_imp_sv_2: fixes R :: "('a \<times> 'b) set" assumes "bijective R" shows "single_valued (R\<inverse>)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (A_B.Dom (fst Fg) f) = A.arr f"
fmmap_def: shows "fmmap \<equiv> \<lambda>f. Abs_fmap \<circ> (\<circ>) (map_option f) \<circ> fmlookup"
extract_rpath_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option" shows "Graph.extract_rpath c src dst PRED \<equiv> while\<^sub>T\<^bsup>\<lambda>(v, p). v \<in> dom PRED \<and> Graph.isPath c v p dst \<and> distinct (Graph.pathVertices v p) \<and> (\<forall>v'\<in>set (Graph.pathVertices v p). Graph.min_dist c src v \<le> Graph.min_dist c src v') \<and> Graph.min_dist c src v + length p = Graph.min_dist c src dst\<^esup> (\<lambda>(v, p). v \<noteq> src) (\<lambda>(v, p). assert (v \<in> dom PRED) \<bind> (\<lambda>_. let u = the (PRED v); p = (u, v) # p; v = u in return (v, p))) (dst, []) \<bind> (\<lambda>(uu_, p). return p)"
t_def: shows "t = (\<lambda>i. good_time (B i))"
f4_red_aux_phull_reducible: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and f :: "'t \<Rightarrow>\<^sub>0 'd" assumes "set ps \<subseteq> set bs \<times> set bs"   and "f \<in> phull (set (pdata_pairs_to_list ps))" shows "(red (fst ` set bs \<union> set (f4_red_aux bs ps)))\<^sup>*\<^sup>* f 0"
wf_mstate_msteps: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate"   and v :: "'a option list"   and \<pi>' :: "(char list \<times> 'a list) prefix"   and X :: "(nat \<times> 'a option list) set \<times> 'a mstate"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "mem_restr R v"   and "\<pi> \<le> \<pi>'"   and "X = msteps (pdrop (plen \<pi>) \<pi>') st" shows "wf_mstate \<phi> \<pi>' R (snd X) \<and> ((i, v) \<in> fst X) = ((i, v) \<in> M \<pi>' - M \<pi>)"
of_nat_inject_hmset: fixes m :: "nat"   and n :: "nat" shows "(of_nat m = of_nat n) = (m = n)"
is_hom_GB_bound_def: fixes F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and b :: "nat" shows "extended_ord.is_hom_GB_bound F b = ((\<forall>f\<in>F. homogeneous f) \<longrightarrow> (\<forall>g\<in>extended_ord.punit.reduced_GB F. poly_deg g \<le> b))"
frechet_derivative_eq_vector_derivative: fixes f :: "real \<Rightarrow> 'c"   and x :: "real" assumes "f differentiable at x" shows "frechet_derivative f (at x) = (\<lambda>r. r *\<^sub>R vector_derivative f (at x))"
Compl_iff: fixes c :: "'e"   and A :: "'e set" shows "(c \<in> - A) = (c \<notin> A)"
intro: fixes \<sigma> :: "nat \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and init :: "'d rexp \<Rightarrow> 'e"   and delta :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and final :: "'e \<Rightarrow> bool"   and wf_state :: "'e \<Rightarrow> bool"   and L :: "'e \<Rightarrow> 'c list set"   and n :: "nat" assumes "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state id L n" shows "rexp_DA_no_post \<sigma> wf_atom project lookup init delta final wf_state L n"
wuntil_def: fixes \<gamma>' :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma>' \<WW>\<^sub>b \<gamma> \<equiv> \<gamma>' \<UU>\<^sub>b \<gamma> \<or>\<^sup>b \<box>\<^sub>b\<gamma>'"
cut_valid: fixes G :: "prog"   and A' :: "'b triple set"   and ts :: "'c triple set"   and A :: "'d triple set" assumes "G,A'|\<Turnstile>ts"   and "G,A|\<Turnstile>A'" shows "G,A|\<Turnstile>ts"
inject: fixes x1 :: "'e"   and x2 :: "'f"   and y1 :: "'e"   and y2 :: "'f" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
inf_sup_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<squnion> z) \<sqinter> x = y \<sqinter> x \<squnion> z \<sqinter> x"
hcomp_ide_isomorphic: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.isomorphic g h"   and "Span.src f = Span.trg g" shows "Span.isomorphic (f \<circ> g) (f \<circ> h)"
in_set_zip: fixes p :: "'k \<times> 'l"   and xs :: "'k list"   and ys :: "'l list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
equal_double_def: fixes x :: "double"   and y :: "double" shows "equal_class.equal x y = (lcompare_double x y = 0)"
bind_commute_spmf: fixes p :: "'e spmf"   and q :: "'f spmf"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
word_less_def: fixes a :: "'b word"   and b :: "'b word" shows "(a < b) = (uint a < uint b)"
if_p_then_p: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<phi> in v]"
prefix_of_imp_linear: fixes \<pi> :: "'c prefix"   and \<sigma> :: "'c trace"   and \<pi>' :: "'c prefix" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi>' \<sigma>" shows "\<pi> \<le> \<pi>' \<or> \<pi>' \<le> \<pi>"
"2_1": shows "w ||| k ## sscan (count cs) w k = ys @- x ## zs"
"1_3": shows "x = Inf (PLS_charles s as PROB)"
hypreal_sqrt_gt_zero: fixes x :: "real star" assumes "0 < x" shows "0 < (*f* sqrt) x"
prems_1: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"
DynPDG_ddep_edge_no_shorter_ddep_edge: fixes n :: "'node"   and V :: "'var"   and as :: "'edge list"   and n' :: "'node" assumes "n -{V}as\<rightarrow>\<^sub>d\<^sub>d n'" shows "\<forall>as' a as''. tl as = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'"
fresh_var_subst: fixes e :: "exp"   and x :: "Vars.var" shows "e[fresh_var e::=x] = e"
i_take_first: fixes f :: "nat \<Rightarrow> 'd" shows "f \<Down> Suc 0 = [f 0]"
ordLeq_refl_on: shows "refl_on {r. Well_order r} ordLeq"
intro: fixes abind :: "'a \<Rightarrow> (unit \<Rightarrow> 'b) \<Rightarrow> 'b"   and areturn :: "unit \<Rightarrow> 'a"   and aASSERT :: "bool \<Rightarrow> 'a"   and aASSUME :: "bool \<Rightarrow> 'a" assumes "generic_Assert abind areturn aASSERT aASSUME" shows "transfer_generic_Assert_remove abind areturn aASSERT aASSUME"
cfun_meet_simp: fixes f :: "'b \<rightarrow> 'a"   and g :: "'b \<rightarrow> 'a"   and x :: "'b" shows "(f \<squnion> g)\<cdot>x = f\<cdot>x \<squnion> g\<cdot>x"
B: assumes "\<forall>u\<in>U. (u, D x) \<notin> I" shows "ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr> \<in> ipurge_fail_aux_t_set \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr>"
i_2: shows "n div 2 < 2 ^ (i + 1)"
COMP_def: fixes g :: "nat list \<Rightarrow> nat"   and fs :: "(nat list \<Rightarrow> nat) list"   and l :: "nat list" shows "COMP g fs l = g (map (\<lambda>f. f l) fs)"
st: shows "(if fc = 0 then state else upd_fi_mu_state state i (vec n (\<lambda>i. fi_state state $ i - fc * fj $ i)) (IArray.of_fun (\<lambda>jj. if jj < j - 1 then dmu_ij_state state i jj - fc * dmu_ij_state state (j - 1) jj else if jj = j - 1 then dmu_ij_state state i jj - d_state state j * fc else dmu_ij_state state i jj) i)) = st"
zero_sheaf_spec_in_sheaf_spec: fixes U :: "('a \<times> 'a) set set set" assumes "local.local.is_zariski_open U" shows "local.local.zero_sheaf_spec U \<in> local.local.sheaf_spec U"
induct: fixes P :: "trilean \<Rightarrow> trilean \<Rightarrow> bool"   and a0 :: "trilean"   and a1 :: "trilean" assumes "\<And>uu_. P invalid uu_"   and "P true invalid"   and "P false invalid"   and "P true true"   and "P true false"   and "P false true"   and "P false false" shows "P a0 a1"
sset_cycle: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "sset (cycle xs) = set xs"
lookup_pair_eq_value: fixes xs :: "('a \<times> 'c) list"   and v :: "'c"   and k :: "'a" assumes "oalist_inv_raw xs"   and "v \<noteq> (0::'c)" shows "(lookup_pair xs k = v) = ((k, v) \<in> set xs)"
1: shows "local.Left a' = Left_a.comp \<and> local.Right a' = Right_a.comp"
return_pmf_inj: fixes x :: "'c"   and y :: "'c" shows "(return_pmf x = return_pmf y) = (x = y)"
game_sem_loop: fixes I :: "interp"   and \<alpha> :: "game" shows "game_sem I \<alpha>** = (\<lambda>X. lfp (\<lambda>Z. X \<union> game_sem I \<alpha> Z))"
deg: shows "\<forall>r\<in>set rs. degree r \<le> 2"
pCons_1: assumes "degree p2 < degree p1" shows "fcompose (p1 + p2) q1 q2 = fcompose p1 q1 q2 + q2 ^ (degree p1 - degree p2) * fcompose p2 q1 q2"
\<pi>_idempotent: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "\<pi> k T (\<pi> k T I) = \<pi> k T I"
rel_gc_def: fixes gc :: "('c, 'ls, 'gs) global_config"   and gd :: "('d, 'ls, 'gs) global_config" shows "rel_gc gc gd \<equiv> rel_mset rel_lc (processes gc) (processes gd) \<and> global_config.state gc = global_config.state gd"
UNION_eq: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "\<mu> (\<P> B A) = {y. \<exists>x\<in>A. s2p (B x) y}"
val_ifex_ite: fixes fi :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and i :: "'a ifex"   and ft :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and t :: "'a ifex"   and fe :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and e :: "'a ifex"   and ass :: "'a \<Rightarrow> bool" assumes "\<And>ass. fi ass = val_ifex i ass"   and "\<And>ass. ft ass = val_ifex t ass"   and "\<And>ass. fe ass = val_ifex e ass"   and "ifex_ordered i"   and "ifex_ordered t"   and "ifex_ordered e" shows "bf_ite fi ft fe ass = val_ifex (ifex_ite i t e) ass"
finsum_Un_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a vec" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier_vec n"   and "g \<in> B \<rightarrow> carrier_vec n" shows "finsum V g (A \<union> B) = finsum V g A + finsum V g B"
assms: fixes x :: "'a ltl"   and y :: "'a ltl" shows "propos (f x y) = propos x \<union> propos y"
homologous_rel_def: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "homologous_rel p X S \<equiv> \<lambda>a b. singular_relboundary p X S (a - b)"
lin_indpt_full_rank: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "distinct (cols A)"   and "lin_indpt (set (cols A))" shows "rank A = nc"
AB'_disj: shows "comp_GSMP_disjoint public arity Ana \<Gamma> A' B' A B C"
simps_2: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "singleCombinators (x \<oplus> y # xs) = False"
distributed_swap: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e"   and Pxy :: "'d \<times> 'e \<Rightarrow> ennreal" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "distributed M (T \<Otimes>\<^sub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
dvd_smult_int: fixes c :: "int"   and q :: "int poly"   and p :: "int poly" assumes "c \<noteq> 0"   and "q dvd Polynomial.smult c p" shows "primitive_part q dvd p"
I3: assumes "Can v" shows "\<lbrace>Inv v\<rbrace> = local.inv \<lbrace>v\<rbrace>"
set_intros: fixes x2 :: "'a" shows "x2 \<in> set_extended \<lbrakk>x2\<rbrakk>"
2: shows "(f \<sqinter> - q)\<^sup>\<star> * (f \<sqinter> q)\<^sup>T = (f \<sqinter> q)\<^sup>T"
vsv_vlrestriction: fixes r :: "V"   and A :: "V" assumes "vsv r" shows "vsv (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xs :: "'a list" shows "groupF f (x # xs) = (x # filter (\<lambda>y. f x = f y) xs) # groupF f (filter (\<lambda>y. f x \<noteq> f y) xs)"
IH: fixes x :: "'a" assumes "pairwise coprime (set ys)"   and "distinct ys"   and "\<And>y. y \<in> set ys \<Longrightarrow> is_unit (lift y)" shows "(\<Sum>i<length ys. lift (decompose x ys ! i) div lift (ys ! i)) = lift x div lift (prod_list ys)"
nth_Cons_pos: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" assumes "0 < n" shows "(x # xs) ! n = xs ! (n - 1)"
meval_sched_snocI: fixes ns :: "nat list"   and cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1'' :: "'Var \<Rightarrow> 'Val"   and n :: "nat"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'', mem\<^sub>1'') \<and> (cms\<^sub>1'', mem\<^sub>1'') \<leadsto>\<^bsub>n\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns @ [n]\<^esub> (cms\<^sub>1', mem\<^sub>1')"
varNotIn_degree: fixes var :: "nat"   and p :: "'a mpoly" assumes "var \<notin> vars p" shows "MPoly_Type.degree p var = 0"
I1: shows "blinfun_scaleR_left (inverse (blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)))) o\<^sub>L (Ds (flow0 (fst (x, t)) (snd (x, t))) o\<^sub>L flowderiv (fst (x, t)) (snd (x, t)) o\<^sub>L embed2_blinfun) = 1\<^sub>L"
in_set_tlD: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
sterm_induct: fixes P :: "sterm \<Rightarrow> bool"   and t :: "sterm" assumes "\<And>x. P (Sconst x)"   and "\<And>x. P (Svar x)"   and "\<And>cs. (\<And>pat t. (pat, t) \<in> set cs \<Longrightarrow> P t) \<Longrightarrow> P (Sabs cs)"   and "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (t $\<^sub>s u)" shows "P t"
borel_measurable_count_space: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "f \<in> borel_measurable (count_space S)"
spray_event: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.spray \<P> x \<noteq> {}" shows "x \<in> \<E>"
prems: shows "lfilter P xs = lappend (LCons y ys) zs"
norm_unboxed_bind_OpDyn: shows "Subx.norm_unboxed \<circ> OpDyn = id"
c: shows "h \<turnstile> local.a_get_scdom_component ptr \<rightarrow>\<^sub>r c"
ptrm_def: fixes n :: "nat" shows "ptrm n \<equiv> {\<sigma> \<in> fmla. Fvars \<sigma> = Variable ` {0..n} \<and> prv (exu out \<sigma>)}"
concat_eq_appendD: fixes xss :: "'a list list"   and ys :: "'a list"   and zs :: "'a list" assumes "concat xss = ys @ zs"   and "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2"
RPHIL_rec: fixes i :: "nat" shows "RPHIL i = (picks i i \<rightarrow> picks i (i - 1) \<rightarrow> putsdown i (i - 1) \<rightarrow> putsdown i i \<rightarrow> RPHIL i)"
state_transition_refl_def: fixes s :: "'a"   and s' :: "'a" shows "s \<rightarrow>\<^sub>i* s' \<equiv> (s, s') \<in> {(x, y). x \<rightarrow>\<^sub>i y}\<^sup>*"
simps_2: fixes vb :: "String.literal"   and vc :: "typ"   and va :: "Core.term list" shows "strip_comb_imp (Ct vb vc, va) = (Ct vb vc, va)"
gcd_impl_def: fixes f :: "'a poly"   and g :: "'a poly" shows "gcd_impl f g = (if length (coeffs g) \<le> length (coeffs f) then gcd_impl_main f g else gcd_impl_main g f)"
rebase_id: fixes x :: "'b mod_ring" shows "@@x = x"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "srev\<cdot>(x :# xs) = srev\<cdot>xs :@ [:x:]"
Times_mset_single_right: fixes A :: "'a multiset"   and b :: "'b" shows "A \<times># {#b#} = {#(a, b). a \<in># A#}"
FF_def: shows "\<tau>.G.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (G (fst \<mu>\<nu>), G (snd \<mu>\<nu>)) else C.VV.null"
linear_add: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and b1 :: "('c, 'd) vec"   and b2 :: "('c, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (b1 + b2) = f b1 + f b2"
finite_combination_1: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option" assumes "finite (RID\<^sub>S \<sigma>)"   and "finite (RID\<^sub>S \<tau>)" shows "finite (RID\<^sub>S (\<sigma>;;\<tau>))"
poly_higher_pderiv_aux2: fixes n :: "nat"   and c :: "'a"   and p :: "'a poly" shows "poly ((pderiv ^^ n) ([:- c, 1::'a:] ^ n * p)) c = fact n * poly p c"
deriv_upwards: fixes n :: "nat"   and list :: "(nat \<times> form) list"   and s :: "(nat \<times> form) list" assumes "(n, list) \<in> deriv s"   and "\<not> is_axiom (s_of_ns list)" shows "\<exists>zs. (Suc n, zs) \<in> deriv s \<and> zs \<in> set (subs list)"