hcju/isabelleps huggingface dataset:
l_new_element_get_tag_name_axioms_def: fixes get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_tag_name_axioms get_tag_name get_tag_name_locs \<equiv> (\<forall>ptr' new_element_ptr h h' r. ptr' \<noteq> new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_element_ptr h'. h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_tag_name new_element_ptr \<rightarrow>\<^sub>r [])"
frees_free: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "frees (free name) = {|name|}"
fds_scale_add: fixes c :: "nat"   and f :: "'b fds"   and g :: "'b fds" shows "fds_scale c (f + g) = fds_scale c f + fds_scale c g"
cons_cluster_has_ne_slices: fixes C :: "'node set"   and p :: "'node"   and Sl :: "'node set" assumes "is_cons_cluster C"   and "p \<in> C"   and "Sl \<in> slices p" shows "Sl \<noteq> {}"
prems: shows "wbt \<langle>l, (a, b_), r\<rangle>"
norm_eq_sqrt_cinner: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "norm x = sqrt (cmod (cinner x x))"
gFreshGAbs2_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshGAbs2 MOD \<equiv> \<forall>ys y xs x s X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> fresh ys y X' \<and> gFresh MOD ys y X' X \<longrightarrow> gFreshAbs MOD ys y (Abs xs x X') (gAbs MOD xs x X' X)"
inv_is_inverse: fixes f :: "'a2" assumes "C2.iso f" shows "C2.inverse_arrows f (C2.inv f)"
grid_base_union: fixes b :: "(nat \<times> int) list"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and ds :: "nat set"   and x :: "(nat \<times> int) list"   and ds' :: "nat set" assumes "b \<in> sparsegrid' dm"   and "p \<in> grid (base ds b) ds"   and "x \<in> grid (base ds' p) ds'" shows "x \<in> grid (base (ds \<union> ds') b) (ds \<union> ds')"
float_pos_eq_mantissa_pos: fixes x :: "float" shows "(0 < x) = (0 < mantissa x)"
pre_ranks_Min_in: fixes n :: "nat"   and \<nu> :: "'a"   and q :: "'b" assumes "pre_ranks (\<lambda>q. state_rank q n) \<nu> q \<noteq> {}" shows "Min (pre_ranks (\<lambda>q. state_rank q n) \<nu> q) \<in> pre_ranks (\<lambda>q. state_rank q n) \<nu> q"
sel_6: fixes x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" shows "arg (App x21 x22) = x22"
bisimulation_final_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" assumes "bisimulation_final trsys1 trsys2 bisim tlsim final1 final2" shows "bisimulation_final trsys2 trsys1 (flip bisim) (flip tlsim) final2 final1"
uminus_mat_def: fixes A :: "'b mat" shows "- A \<equiv> mat (dim_row A) (dim_col A) (\<lambda>ij. - A $$ ij)"
partial_order_on_converse: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "partial_order_on A (r\<inverse>) = partial_order_on A r"
is_path_undir_split: fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_path_undir G v (p1 @ p2) v' = (\<exists>u. is_path_undir G v p1 u \<and> is_path_undir G u p2 v')"
one_dim_iso_times: fixes \<psi> :: "'c"   and \<phi> :: "'c" shows "one_dim_iso (\<psi> * \<phi>) = one_dim_iso \<psi> * one_dim_iso \<phi>"
algebraic_csqrt_iff: fixes x :: "complex" shows "algebraic (csqrt x) = algebraic x"
boundOutputEqSupp: fixes x :: "name"   and B :: "('d, 'e, 'f) boundOutput"   and y :: "name"   and C :: "('d, 'e, 'f) boundOutput" assumes "\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>y\<rparr>C"   and "x \<in> supp B" shows "y \<in> supp C"
result_DONE: fixes a :: "'f" shows "Resumption.resumption.result (DONE a) = Some a"
assms_3: shows "AE x in M. h x \<noteq> \<infinity>"
list_all_zip_alt: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and as :: "'a list"   and bs :: "'b list" shows "list_all_zip P as bs = (length as = length bs \<and> (\<forall>i<length as. P (as ! i) (bs ! i)))"
a_setmult_rcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier R"   and "K \<subseteq> carrier R"   and "x \<in> carrier R" shows "H <+> (K +> x) = H <+> K +> x"
vsum_0_eqpoll: fixes a :: "V" shows "elts (0 \<Uplus> a) \<approx> elts a"
record_simps_2: fixes f :: "'h \<Rightarrow> 'g"   and x :: "('h, 'f) foo" shows "foo.field_2 (foo.update_field_1 f x) = foo.field_2 x"
prime_divisors: fixes p :: "nat" shows "prime p = (divisors p = {1, p} \<and> 1 < p)"
blinfun_eqI: fixes x :: "'e \<Rightarrow>\<^sub>L 'f"   and y :: "'e \<Rightarrow>\<^sub>L 'f" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst'': fixes t :: "('c, 'd) Term.term"   and b :: "('c, 'd) stateful_strand_step"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)" shows "\<exists>s\<in>trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p b. t = s \<cdot> rm_vars (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b)) \<delta>"
apps_betasE: fixes r :: "lam"   and rs :: "lam list"   and s :: "lam"   and x :: "name" assumes "r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> s" obtains r' :: "lam" where "r \<rightarrow>\<^sub>\<beta> r'" and "s = r' \<degree>\<degree> rs"   | rs' :: "lam list" where "rs [\<rightarrow>\<^sub>\<beta>]\<^sub>1 rs'" and "s = r \<degree>\<degree> rs'"   | t :: "lam" and u :: "lam" and us :: "lam list" where "x \<sharp> r \<Longrightarrow> r = (Lam [x].t) \<and> rs = u # us \<and> s = t[x::=u] \<degree>\<degree> us"
apply_sequence: fixes f :: "'b perm"   and g :: "'b perm"   and a :: "'b" shows "apply f (apply g a) = apply (f * g) a"
normal_9: shows "s' = (redT_updLs (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, (redT_updTs (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', redT_updLns (locks s) t no_wait_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>)), m'), ws'_, redT_updIs (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
map_scale_zero_left: fixes t :: "'c \<Rightarrow>\<^sub>0 'd" shows "(0::'d) \<cdot> t = 0"
Close2: fixes P :: "pi"   and a :: "name"   and y :: "name"   and P' :: "pi"   and Q :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>l a<\<nu>y> \<prec>\<^sub>l P'"   and "Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'"   and "y \<sharp> Q" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l <\<nu>y>(P' \<parallel> Q'[x::=y])"
d\<^sub>0_def: shows "Semantics.d\<^sub>0 \<equiv> map_fun eval\<o> id Some"
path_image_circlepath_minus: fixes z :: "complex"   and r :: "real" shows "path_image (circlepath z (- r)) = path_image (circlepath z r)"
less_diff_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (minus c b) = less (plus a b) c"
lens_plus_def: fixes X :: "'b \<Longrightarrow> 'd"   and Y :: "'c \<Longrightarrow> 'd" shows "X +\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. (get\<^bsub>X\<^esub> \<sigma>, get\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> (u, v). put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u\<rparr>"
distinct_211: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x16 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Unauth x16"
lim_mat_A: shows "limit_mat f (mat dim dim (\<lambda>(i, j). lim (\<lambda>n. f n $$ (i, j)))) dim"
assms: shows "supp (map fst \<theta>) \<sharp>* u"
subspace_orthogonal_to_vectors: fixes S :: "'a set" shows "subspace {y. \<forall>x\<in>S. orthogonal x y}"
preprm_ext_reflexive: fixes x :: "('b \<times> 'b) list" shows "x =p x"
convex_cone: fixes C :: "'a vec set" assumes "C \<subseteq> carrier_vec n" shows "convex (cone C)"
eq_rowI: fixes B :: "'a mat"   and A :: "'a mat" assumes "\<And>i. i < dim_row B \<Longrightarrow> row A i = row B i"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
exhaust: fixes y :: "real_alg_2" obtains x1 :: "rat" where "y = Rational x1"   | x21 :: "nat" and x22 :: "int poly \<times> rat \<times> rat" where "y = Irrational x21 x22"
exp_add_not_zero_imp_right: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m + n) \<noteq> zero" shows "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero"
lift_subst_bv2_subst_lt: fixes i :: "nat"   and j :: "nat"   and t :: "Core.term"   and s :: "Core.term" assumes "i < j + 1" shows "lift (subst_bv2 t j s) i = subst_bv2 (lift t i) (j + 1) (lift s i)"
leq_zero: fixes x :: "'a" assumes "Field r \<noteq> {}" shows "((x, r.zero) \<in> r) = (x = r.zero)"
of_bits_drop: fixes n :: "nat"   and bs :: "bool list" assumes "n < length bs" shows "of_bits (drop n bs) = drop_bit n (of_bits bs)"
assms_2: shows "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"
compactin_def: fixes X :: "'c topology"   and S :: "'c set" shows "compactin X S = (S \<subseteq> topspace X \<and> (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> S \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union> \<F>)))"
constant_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" shows "constant_functor A B b \<equiv> category A \<and> category B \<and> constant_functor_axioms B b"
psubstAbs_idEnv: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A" shows "A $[idEnv] = A"
intros_6: fixes cl :: "cl"   and t1s :: "t list"   and t2s :: "t list"   and f :: "host"   and ves :: "e list"   and vcs :: "v list"   and n :: "nat"   and m :: "nat"   and s :: "s"   and hs :: "host_state"   and s' :: "s"   and vcs' :: "v list"   and vs :: "v list"   and i :: "nat" assumes "cl = Func_host (t1s _> t2s) f"   and "ves = $$* vcs"   and "length vcs = n"   and "length t1s = n"   and "length t2s = m"   and "host_apply s (t1s _> t2s) f vcs hs = Some (s', vcs')" shows "\<lparr>s;vs;ves @ [Callcl cl]\<rparr> \<leadsto>_ i \<lparr>s';vs;$$* vcs'\<rparr>"
af_F: fixes \<phi> :: "'a ltl"   and w :: "'a set list" shows "af (F \<phi>) w = Or (F \<phi> # map (af \<phi>) (suffixes w))"
diagonal_tendsto: fixes n :: "nat" shows "\<exists>l. (\<lambda>i. (fm (Suc (diagseq i)) (y (Suc (diagseq i))))\<^sub>F n) \<longlonglongrightarrow> l"
and_foldl_sem: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and \<Gamma> :: "('b, 'c, 'a) formula list"   and \<phi> :: "('b, 'c, 'a) formula" assumes "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT)"   and "List.member \<Gamma> \<phi>" shows "\<nu> \<in> fml_sem I \<phi>"
sorted_list_of_set_unique: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and l :: "'a list" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.strict_sorted less l \<and> list.set l = A \<and> length l = card A) = (linorder.sorted_list_of_set less_eq A = l)"
lt: shows "component_of_term (lt p) = component_of_term v"
Reach_def: shows "Reach = reach \<Sigma> (\<delta>\<^sub>m k) (the (\<iota>\<^sub>m k)) \<times> reach \<Sigma> (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) (\<iota>\<^sub>m(k := None))"
fin: shows "finite {x. r < x \<and> poly [:c, b, a:] x = 0}"
times_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<odot> g = (\<lambda>(i, j). sup_monoid.sum (\<lambda>k. f (i, k) * g (k, j)) {k. True})"
red_no_deadlock: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -1-t\<triangleright>ta\<rightarrow> s'"   and "t \<in> r1.deadlocked s" shows "False"
lhs_linear_poly_to_eq: fixes h :: "linear_poly"   and v :: "nat" shows "lhs (linear_poly_to_eq h v) = v"
det_state_cover_empty: fixes M :: "('e, 'f, 'g) FSM"   and V :: "'e list set" assumes "is_det_state_cover M V" shows "[] \<in> V"
assms_3: shows "Cond Rel \<and> Rel \<subseteq> RelA \<and> Rel \<subseteq> RelB"
prems_1: shows "subjective a \<^bold>C\<^bsub>as\<^esub> \<psi>"
FullStruct_axioms: shows "FFullStruct InfModel.intTF local.M.intFF local.M.intPF"
permutations_of_multisetI: fixes xs :: "'a list"   and A :: "'a multiset" assumes "mset xs = A" shows "xs \<in> permutations_of_multiset A"
fold_append: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list" shows "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
case_1: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and x1 :: "'h" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
quotient_identity: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) // skip = c"
irreducible_hom_imp_irreducible: fixes a :: "'a" assumes "irreducible (hom a)" shows "irreducible a"
assms_3: shows "M' = prefixBeforeLastDecision M @ [(opposite (lastDecision M), False)]"
axioms_1: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "pnop A n f" shows "vsv f"
preserves_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>D (F \<mu>) = F.map\<^sub>0 (trg\<^sub>C \<mu>)"
simple_bochner_integral_minus: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" shows "simple_bochner_integral M (\<lambda>x. - f x) = - simple_bochner_integral M f"
matches_DeMorgan: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchNot (MatchAnd m1 m2)) a p = (matches \<gamma> (MatchNot m1) a p \<or> matches \<gamma> (MatchNot m2) a p)"
ZFTSnd_def: shows "ZFTSnd = Snd \<circ> Fst"
bsingleton_def: fixes x :: "'c" shows "bsingleton x = binsert x bempty"
t1_space: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
less_eq_st_rep_iff_fun: fixes r1 :: "(char list \<times> 'a) list"   and r2 :: "(char list \<times> 'a) list" shows "less_eq_st_rep r1 r2 = (fun_rep r1 \<le> fun_rep r2)"
count_image_mset: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e multiset"   and x :: "'d" shows "count (image_mset f A) x = sum (count A) (f -` {x} \<inter> set_mset A)"
someI_ex: fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
\<tau>move1LAssRed: fixes P :: "'a prog"   and h :: "'heap"   and V :: "'b"   and v :: "'addr val" shows "\<tau>move1 P h (V:=Val v)"
strategy_conforming_path_exists_single: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v0 \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>" obtains P :: "'c llist" where "vmc_path G P v0 p \<sigma>"
prem: shows "2 ^ Suc n * q < 2 ^ m"
simps: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list" shows "lhs q \<rightarrow> l qs = q"
harry_sum_closed_form_aux: fixes n :: "nat" shows "4 * harry_sum n = n * (n + 1) * 2 ^ n"
simps_3: shows "X3d X3.Xd2 = D2"
multiplicity_eq_Max: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p x = Max {n. dvd.dvd times (power.power one times p n) x}"
assms: shows "k < length (trace_parallel_plan_sas_plus I \<psi>)"
additive_def: fixes f :: "'a \<Rightarrow> 'a" shows "sup.additive (\<sqinter>) f \<equiv> \<forall>x y. f (x \<sqinter> y) = f x \<sqinter> f y"
rec_c: shows "dmu_array_cost dmus' (Suc i) = (res, c_rec)"
isCont_char: fixes t :: "real" shows "isCont (char M) t"
dotprod_le_left: fixes v :: "nat list"   and w :: "nat list"   and b :: "nat list" assumes "v \<le>\<^sub>v w"   and "length b = length w" shows "v \<bullet> b \<le> w \<bullet> b"
map_up: fixes a :: "'a"   and b :: "'b list"   and m :: "'m" shows "map_of (up a b m) = (map_of m)(a := b)"
stutter_reduced_reduced: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<natural>\<natural>\<sigma> = \<natural>\<sigma>"
esqrt_simps_3: shows "esqrt \<infinity> = \<infinity>"
continuous_at_winding_number: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "isCont (winding_number \<gamma>) z"
multiset_of_subseqs: fixes xs :: "'a list" shows "mset ` set (subseqs xs) = {ps. ps \<subseteq># mset xs}"
square_mod_existence: fixes n :: "nat"   and k :: "nat" assumes "\<exists>q. q\<^sup>2 = n" shows "\<exists>q. n mod k = q\<^sup>2 mod k"
concat_append: fixes xs :: "'a list list"   and ys :: "'a list list" shows "concat (xs @ ys) = concat xs @ concat ys"
WTrt_elim_cases_15: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e\<^sub>1 :: "'addr expr"   and bop :: "bop"   and e\<^sub>2 :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 : T" obtains T1 :: "ty" and T2 :: "ty" where "P,E,h \<turnstile> e\<^sub>1 : T1" and "P,E,h \<turnstile> e\<^sub>2 : T2" and "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T"
numeral_times_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "num"   and n :: "num" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m * n)"
leao: fixes w' :: "'a word"   and x' :: "'a word"   and y' :: "'a word" assumes "w' = x' AND y'" shows "x' = x' OR w'"
\<beta>_boundedness_diag_eq: fixes y :: "'a"   and m :: "int"   and x :: "'a" assumes "- int (k y) \<le> m"   and "m \<le> int (k x)"   and "x \<in> X"   and "y \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x - u y = real_of_int m}"
inj_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
cdf_diff_eq: fixes x :: "real"   and y :: "real"   and k :: "nat" assumes "x < y" shows "cdf (\<mu> k) y - cdf (\<mu> k) x = \<mu>.prob k {x<..y}"
CatIdCompId: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X = id\<^bsub>C\<^esub> X"
fresh_vsubst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X" shows "fresh zs z (X #[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
"1_1": shows "fmdom' (fmrestrict_set {v \<in> fmdom' s1. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s1 v} s1) = {v \<in> fmdom' s1. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s1 v}"
a_rcos_zero: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set"   and i :: "'e" assumes "ring R"   and "ideal I R"   and "i \<in> I" shows "I +>\<^bsub>R\<^esub> i = I"
dom_eq_singleton_conv: fixes f :: "'d \<Rightarrow> 'e option"   and x :: "'d" shows "(dom f = {x}) = (\<exists>v. f = [x \<mapsto> v])"
nstd_case1: fixes a :: "nat"   and b :: "cell list"   and c :: "cell list" assumes "0 < a" shows "NSTD (trpl_code (a, b, c))"
length_Tensor: fixes M1 :: "int poly fract list list"   and M2 :: "int poly fract list list" shows "length (M1 \<otimes> M2) = length M1 *n length M2"
\<pi>_defined: fixes u :: "'v" assumes "Q' Q \<pi> u u \<noteq> \<infinity>" shows "\<pi>' Q \<pi> u u \<noteq> None"
wens_set_subset_single: fixes act :: "('b \<times> 'b) set"   and init :: "'b set"   and allowed :: "('b \<times> 'b) set set"   and B :: "'b set" assumes "single_valued act" shows "wens_set (mk_program (init, {act}, allowed)) B \<subseteq> insert (wens_single act B) (range (wens_single_finite act B))"
resume_bind_resumption: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done (x \<bind> f)" shows "Resumption.resumption.resume (x \<bind> f) = (if Resumption.resumption.is_Done x then Resumption.resumption.resume (f (the (Resumption.resumption.result x))) else (\<lambda>inp. Resumption.resumption.resume x inp \<bind> f))"
bl_bin_bl: fixes bs :: "bool list" shows "bin_to_bl (length bs) (bl_to_bin bs) = bs"
eq: shows "[(a, c)] \<bullet> x = [(b, c)] \<bullet> y"
Theorem_11_fp_cop_F: shows "\<exists>y\<in>TruePref.cop ds. (x, y) \<in> Pd' (Xd x)"
phis_phi: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "Mapping.lookup (phis g) (n, v) = Some vs" shows "uninst_code.ssa.phi phis g v = Some vs"
compile: shows "rec_ci (Mn n f) = (ap, ar, ft) \<and> length args = ar"
step: shows "S t j \<turnstile> t ! j \<mapsto> S t (j + 1)"