hcju/isabelleps huggingface dataset:
QP: shows "(expand_init < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming (fst (n, ns)). nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd (n, ns)) \<and> ((\<forall>nm\<in>incoming (fst (n, ns)). nm < name (fst (n, ns))) \<and> 0 < name (fst (n, ns)) \<and> (\<forall>q\<in>snd (n, ns). name q < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming q. nm < name (fst (n, ns))))) \<and> expand_name_ident (snd (n, ns))) \<and> (\<forall>nd\<in>ns. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` ns))"
bij_betw_iff_card: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "(\<exists>f. bij_betw f A B) = (card A = card B)"
wf_cdecl_overrides_SomeD: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and sig :: "sig"   and newM :: "methd"   and old :: "qtname \<times> methd" assumes "wf_cdecl G (C, c)"   and "C \<noteq> Object"   and "table_of (methods c) sig = Some newM"   and "G \<turnstile> qmdecl sig (C, newM) overrides\<^sub>S qmdecl sig old" shows "G\<turnstile>resTy newM\<preceq>resTy old \<and> accmodi old \<le> accmodi newM \<and> \<not> is_static old"
IH: assumes "(a, b, c) \<in> set xs"   and "distinct (map fst xs)" shows "\<exists>pre suf. xs = pre @ (a, b, c) # suf \<and> (\<forall>y\<in>set pre. fst y \<noteq> a)"
ev_4: shows "\<forall>\<^sub>F x in at_right 0. (g0 has_real_derivative g' x) (at x)"
jkbpTn_length: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpTn jkbp envInit envAction envTrans envVal jview n T" shows "tLength t = n"
rep_eq: fixes x :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'d pdevs" shows "pdevs_apply (filter_pdevs x xa) = filter_pdevs_raw x (pdevs_apply xa)"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst'': fixes t :: "('d, 'e) Term.term"   and b :: "('d, 'e) stateful_strand_step"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) 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>"
withdraw_reservation_def: fixes ts :: "traffic"   and c :: "cars"   and n :: "nat"   and ts' :: "traffic" shows "ts \<^bold>\<midarrow>wdr( c, n ) \<^bold>\<rightarrow> ts' \<equiv> pos ts' = pos ts \<and> res ts' = (res ts)(c := Abs_nat_int {n}) \<and> dyn ts' = dyn ts \<and> clm ts' = clm ts \<and> physical_size ts' = physical_size ts \<and> braking_distance ts' = braking_distance ts \<and> n \<^bold>\<in> res ts c \<and> |res ts c| = 2"
norm_pre_pure_rule1: fixes b :: "bool"   and P :: "assn"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" assumes "b \<Longrightarrow> <P> f <Q>" shows "<P * \<up> b> f <Q>"
surjD: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "surj f" shows "\<exists>x. y = f x"
add_power_poly_mod_ring: fixes x :: "'b mod_ring poly"   and y :: "'b mod_ring poly" shows "(x + y) ^ CARD('b) = x ^ CARD('b) + y ^ CARD('b)"
rbb_rfb: shows "bb\<^sub>\<R> = fb\<^sub>\<R> \<circ> \<smile>"
norm_vec_def: fixes x :: "('a, 'b) vec" shows "\<parallel>x\<parallel> = L2_set (\<lambda>i. \<parallel>x $ i\<parallel>) UNIV"
bd_FGl_def: shows "bd_FGl = bd_F *c bd_G +c bd_F"
that: fixes b :: "'a list"   and r :: "'a list" assumes "b \<in> carrier (K [X])"   and "r \<in> carrier (K [X])"   and "long_divides p q (b, r)" shows "thesis"
nat_filtration_from_eq_sets: fixes p :: "real"   and M :: "bool stream measure"   and N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "infinite_coin_toss_space p M"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "sets (infinite_coin_toss_space.nat_filtration N n) = sets (infinite_coin_toss_space.nat_filtration M n)"
next_and: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F \<and> G) = (\<circle>F \<and> \<circle>G)"
induct: fixes x :: "state"   and P :: "state \<Rightarrow> bool" assumes "x \<in> reach"   and "\<And>initk. inj initk \<Longrightarrow> P \<lparr>state.owns = Map.empty, currk = initk, issued = range initk, cards = \<lambda>g. {}, roomk = initk, isin = \<lambda>r. {}, safe = \<lambda>r. True\<rparr>"   and "\<And>s k r g. \<lbrakk>s \<in> reach; P s; k \<notin> state.issued s\<rbrakk> \<Longrightarrow> P (s\<lparr>currk := (state.currk s)(r := k), issued := state.issued s \<union> {k}, cards := (state.cards s)(g := state.cards s g \<union> {(state.currk s r, k)}), owns := (state.owns s)(r := \<lfloor>g\<rfloor>), safe := (state.safe s)(r := False)\<rparr>)"   and "\<And>s k k' g r. \<lbrakk>s \<in> reach; P s; (k, k') \<in> state.cards s g; state.roomk s r \<in> {k, k'}\<rbrakk> \<Longrightarrow> P (s\<lparr>isin := (state.isin s)(r := state.isin s r \<union> {g}), roomk := (state.roomk s)(r := k'), safe := (state.safe s)(r := state.owns s r = \<lfloor>g\<rfloor> \<and> state.isin s r = {} \<and> k' = state.currk s r \<or> state.safe s r)\<rparr>)"   and "\<And>s g r. \<lbrakk>s \<in> reach; P s; g \<in> state.isin s r\<rbrakk> \<Longrightarrow> P (s\<lparr>isin := (state.isin s)(r := state.isin s r - {g})\<rparr>)" shows "P x"
gs_def: shows "gs \<equiv> mset (berlekamp_hensel p n g)"
orderI: fixes a :: "'e"   and b :: "'e" assumes "a = inf a b" shows "a \<le> b"
alt_d_def_nc_nc: fixes x :: "'a" shows "d (x \<sqinter> nc) = 1\<^sub>\<sigma> \<sqinter> (x \<sqinter> nc) \<cdot> 1\<^sub>\<pi> \<parallel> nc"
that_2: fixes i :: "'a" assumes "i \<in> A" shows "[x = y] (mod m i)"
D_nonneg: fixes n :: "nat" shows "0 \<le> D y n"
semiconnex_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "semiconnex A r = semiconnex A r'"
OUTfromChCorrect_data18: shows "OUTfromChCorrect data18"
red_dfs_inv_def: fixes E :: "('b \<times> 'b) set"   and U :: "'b set"   and reds :: "'b set"   and onstack :: "'b set" shows "red_dfs_inv E U reds onstack \<equiv> E `` U \<subseteq> U \<and> finite U \<and> reds \<subseteq> U \<and> E `` reds \<subseteq> reds \<and> E `` reds \<inter> onstack = {}"
sorted_repeat_poly_list_rel_Cons_iff: fixes ys :: "char list list"   and n :: "int"   and p :: "(char list list \<times> int) list"   and a :: "(char list multiset \<times> int) multiset"   and S :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" shows "(((ys, n) # p, a) \<in> sorted_repeat_poly_list_rel S) = ((p, remove1_mset (mset ys, n) a) \<in> sorted_repeat_poly_list_rel S \<and> (mset ys, n) \<in># a \<and> (\<forall>x\<in>set p. S ys (fst x)) \<and> sorted_wrt var_order ys \<and> distinct ys \<and> n \<noteq> 0 \<and> nonzero_coeffs a)"
card_support_1_probability_1: fixes p :: "'b pmf" assumes "card (set_pmf p) = 1" shows "\<forall>e\<in>set_pmf p. pmf p e = 1"
strict_trans2: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a \<prec>\<^sub>p b"   and "b \<preceq>\<^sub>p c" shows "a \<prec>\<^sub>p c"
ax_def: shows "ax \<equiv> solution sys (VAR (Suc 0))"
cong_exp_trans_1: fixes a :: "'a"   and b :: "nat"   and c :: "'a"   and n :: "'a"   and d :: "'a" assumes "[a ^ b = c] (mod n)"   and "[a = d] (mod n)" shows "[d ^ b = c] (mod n)"
fcompose_nzero: fixes p :: "'b poly"   and q2 :: "'b poly"   and q1 :: "'b poly" assumes "p \<noteq> 0"   and "q2 \<noteq> 0"   and "\<forall>c. q1 \<noteq> smult c q2"   and "infinite UNIV" shows "fcompose p q1 q2 \<noteq> 0"
sets_completionE: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets (completion M)" obtains S :: "'b set" and N :: "'b set" and N' :: "'b set" where "A = S \<union> N" and "N \<subseteq> N'" and "N' \<in> null_sets M" and "S \<in> sets M"
dis: shows "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> B j1 \<inter> B j2 = {}"
ang_vec_sym_pi: fixes z1 :: "complex"   and z2 :: "complex" assumes "\<angle> z1 z2 = pi" shows "\<angle> z1 z2 = \<angle> z2 z1"
kim: shows "k ` ({0..1} \<times> S) \<subseteq> sphere 0 1"
non_terminating_is_right: fixes P :: "'b process" shows "non_terminating P = (\<forall>s\<in>T P. tickFree s)"
simps_2: fixes e :: "expr" shows "radicals (Negation e) = radicals e"
that: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and z :: "nat" assumes "rb_aux args = ((bs, ss, []), z)" shows "thesis"
split_list_first_propE: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'a list" and x :: "'a" and zs :: "'a list" where "xs = append ys (Cons x zs)" and "P x" and "\<forall>y\<in>set ys. \<not> P y"
simps_4: fixes v :: "val" shows "throw_of (Val v) = None"
distinct_25: fixes x41 :: "'b formula"   and x42 :: "'b formula"   and x51 :: "'b formula"   and x52 :: "'b formula" shows "x41 \<^bold>\<and> x42 \<noteq> x51 \<^bold>\<or> x52"
valid_Act_fresh: fixes \<alpha> :: "('act, 'pred) S_action"   and P :: "'state"   and x :: "('b, unit, ('act, 'pred) S_action) formula" assumes "bn \<alpha> \<sharp>* P" shows "P \<Turnstile>\<^sub>S Formula.Act \<alpha> x = (\<exists>P'. P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle> \<and> P' \<Turnstile>\<^sub>S x)"
finite_transitivity_chain: fixes A :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "\<And>x. \<not> R x x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"   and "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y" shows "A = {}"
z_in_aug_inf: fixes z :: "int" shows "ant z \<in> Z\<^sub>\<infinity>"
algebraic_plus: fixes x :: "'a"   and y :: "'a" assumes "algebraic x"   and "algebraic y" shows "algebraic (x + y)"
recurrent_subset_infty_returns: fixes x :: "'a"   and A :: "'a set"   and n :: "nat" assumes "x \<in> recurrent_subset_infty A"   and "(T ^^ n) x \<in> A" shows "(T ^^ n) x \<in> recurrent_subset_infty A"
case_3: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f3 :: "nat \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> ((nat \<Rightarrow> nat option) \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f4 :: "'b"   and f5 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x31 :: "nat"   and x32 :: "nat"   and x33 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x34 :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and x35 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x36 :: "nat set"   and x37 :: "nat set"   and x38 :: "nat set"   and x39 :: "nat set" shows "(case RMW x31 x32 x33 x34 x35 x36 x37 x38 x39 of Read x xa xb \<Rightarrow> f1 x xa xb | Write x xa xb xc xd xe xf \<Rightarrow> f2 x xa xb xc xd xe xf | RMW x xa xb xc xd xe xf xg xh \<Rightarrow> f3 x xa xb xc xd xe xf xg xh | Fence \<Rightarrow> f4 | Ghost x xa xb xc \<Rightarrow> f5 x xa xb xc) = f3 x31 x32 x33 x34 x35 x36 x37 x38 x39"
le_supI1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a" shows "x \<le> sup a b"
f_cases: fixes e :: "'v vertex \<times> 'v vertex" obtains x :: "'v" and y :: "'v" where "e = (\<langle>x__\<rangle>, \<langle>y__\<rangle>)"   | y :: "'v" where "e = (SOURCE, \<langle>ya__\<rangle>)"   | x :: "'v" where "e = (\<langle>x\<rangle>, vertex.SINK)"   | y :: "'v" where "e = (vertex.SINK, \<langle>y\<rangle>)"   | "e = (vertex.SINK, vertex.SINK)"   | xo :: "'v vertex" where "e = (xo, SOURCE)"   | "e = (SOURCE, vertex.SINK)"
list_emb_Cons: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and y :: "'c" assumes "list_emb P xs ys" shows "list_emb P xs (y # ys)"
is_solution_def: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "is_solution x A b = (A *v x = b)"
other: fixes g :: "'a"   and l :: "'b"   and g' :: "'a" assumes "(g, l) \<in> oreachable A S U"   and "P (g, l)"   and "U g g'" shows "P (g', l)"
assms_1: shows "f \<in> mor\<^bsub>CatDom (F ;;: G)\<^esub>"
usplit_def: fixes f :: "'b set \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'b set" shows "usplit f A \<equiv> f (A - {Max A}) (Max A)"
2: shows "g = g \<ominus>\<^bsub>UP R\<^esub> h \<oplus>\<^bsub>UP R\<^esub> h"
fmap_rel_nat_the_fmlookup: assumes "(aa, a'a) \<in> \<langle>K, V\<rangle>fmap_rel"   and "(a, a') \<in> K"   and "a' \<in># dom_m a'a" shows "(the (fmlookup aa a), the (fmlookup a'a a')) \<in> V"
mini_less: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "0 < length xs" shows "mini xs key < length xs"
assms_2: fixes s :: "'a" shows "{t. 0 \<le> t} \<subseteq> ll_on_open.existence_ivl UNIV (\<lambda>t. f) UNIV 0 s"
if_eqvt: fixes p :: "perm"   and b :: "bool"   and x :: "'b"   and y :: "'b" shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
A4: shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel"
assms: shows "Fun c [] \<sqsubseteq> t \<cdot> \<sigma>"
disjoint_from_sum: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CEM" shows "(\<not> O z (x \<oplus> y)) = (\<not> O z x \<and> \<not> O z y)"
apply_outputs_register: fixes r :: "nat"   and P :: "vname aexp list"   and p :: "nat"   and i :: "value list"   and c :: "nat \<Rightarrow>f value option" assumes "r < length P" shows "apply_outputs (P[r := V (R p)]) (join_ir i c) ! r = c $ p"
fold_simps_1: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d" shows "fold f [] s = s"
emeasure_eq_measure: fixes p :: "'h spmf"   and A :: "'h set" shows "emeasure (measure_spmf p) A = ennreal (Sigma_Algebra.measure (measure_spmf p) A)"
Finite_Markov_Decision_Process_axioms: shows "Finite_Markov_Decision_Process K S"
assms_3: shows "A, n \<turnstile> ((\<^bold>\<not> (\<^bold>@ i p)) # ps, a) # branch"
r_one: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "x \<otimes> \<one> = x"
d_\<beta>_mono: fixes p :: "fm"   and l :: "int"   and l' :: "int" assumes "iszlfm p"   and "d_\<beta> p l"   and "l dvd l'" shows "d_\<beta> p l'"
cl11: fixes x :: "'a" shows "(x \<sqinter> nc) \<cdot> 1\<^sub>\<pi> \<parallel> nc = (x \<sqinter> nc) \<cdot> nc"
less_ccSup_iff: fixes S :: "'a set" assumes "countable S" shows "(a < Sup S) = (\<exists>x\<in>S. a < x)"
rtrancl_rbt_impl: fixes r :: "('b \<times> 'b) list"   and as :: "'b list" shows "rs.\<alpha> (rtrancl_rbt_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>*}"
assms_4: shows "\<forall>\<^sub>F x in F. f x \<noteq> 0"
sel_1: fixes x1 :: "'a"   and x2 :: "'a tree"   and x3 :: "'a tree" shows "root (Node x1 x2 x3) = x1"
inj_on_image_Pow: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "inj_on ((`) f) (Pow A)"
simps_5: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c = (\<lambda>\<I>. ineq_model \<I> X F \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c \<I>)"
ipoly_mult_rat: fixes r :: "rat"   and p :: "int poly"   and x :: "'a" assumes "r \<noteq> 0" shows "ipoly (poly_mult_rat r p) x = of_int (fst (quotient_of r)) ^ degree p * ipoly p (x * inverse (of_rat r))"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
lookup_except_eq_zeroI: fixes t :: "'c"   and S :: "'c set"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "t \<in> S" shows "lookup (except p S) t = (0::'d)"
induct: fixes P :: "CL \<Rightarrow> bool"   and a0 :: "CL" assumes "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (` t u)"   and "P S"   and "P K"   and "P I" shows "P a0"
apath_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "apath u p v \<equiv> awalk u p v \<and> distinct (awalk_verts u p)"
exhaust: fixes y :: "'a iarray" obtains x :: "'a list" where "y = IArray x"
typedef_bi_unique: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and T :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "bi_unique T"
distinct_4: fixes x3 :: "'f"   and x1 :: "('e, 'f, 'g) Tree set['e]" shows "tPred x3 \<noteq> tConj x1"
closed_w_def: fixes G :: "('c, 'd) pre_digraph"   and p :: "'d list" assumes "wf_digraph G" shows "wf_digraph.closed_w G p \<equiv> \<exists>u. pre_digraph.awalk G u p u \<and> 0 < length p"
surjective: fixes r :: "'a assigned_scheme" shows "r = \<lparr>nrm = nrm r, brk = brk r, \<dots> = assigned.more r\<rparr>"
a2: fixes hbc :: "nat \<Rightarrow> nat"   and dbc :: "nat \<Rightarrow> nat"   and hmining :: "nat \<Rightarrow> bool"   and dmining :: "nat \<Rightarrow> bool"   and sbc :: "nat"   and cnf :: "nat" assumes "bounded_growth hbc dbc hmining dmining sbc cnf" shows "sbc + cnf \<le> hbc 0"
islimpt_approachable: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
inject_2: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list"   and y21 :: "'c"   and y22 :: "('c, 'd) Term.term list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
distinct_10: fixes x61 :: "'b formula"   and x62 :: "'b formula"   and x1 :: "'b" shows "x61 \<^bold>\<rightarrow> x62 \<noteq> Atom x1"
Ana_Fu_cases_1: fixes f :: "('fun, 'atom, 'sets) prot_fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and K :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and g :: "'fun"   and K' :: "(('fun, 'atom, 'sets) prot_fun, nat) Term.term list"   and M' :: "nat list" assumes "Ana (Fun f T) = (K, M)"   and "f = Fu g"   and "Ana\<^sub>f g = (K', M')" shows "(K, M) = (if arity\<^sub>f g = length T \<and> 0 < arity\<^sub>f g then (K' \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (!) T, map ((!) T) M') else ([], []))"
2: shows "state_succ s2 a = snd a ++ s2"
simps: fixes xs :: "'b list" shows "remdups_fwd xs = remdups_fwd_acc {} xs"
above_simps_2: fixes x :: "'a" shows "above (Some x) = {y. x < y}"
capacity_outside: fixes e :: "'v \<times> 'v" assumes "e \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>" shows "capacity (residual_network f) e = 0"
l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr set_child_nodes set_child_nodes_locs"
C5: shows "\<lfloor>mexistsActB Concrete\<rfloor>"
\<alpha>_simp: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "arr f"   and "arr g"   and "arr h" shows "\<a>[f, g, h] = (f \<otimes> g \<otimes> h) \<cdot> \<a>[local.dom f, local.dom g, local.dom h]"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'f, 'g, 'opSym, 'h, 'd, 'e, 'i) model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
homeomorphic_unicoherent_eq: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T" shows "unicoherent S = unicoherent T"
rep_eq: shows "aezfun 1 = (1::'b) \<delta> (0::'a)"
bdd_belowI: fixes A :: "'a set"   and m :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<le> x" shows "bdd_below A"
rank_invar_ins: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.rank_invar bq" shows "BinomialHeapStruc.rank_invar (BinomialHeapStruc.ins t bq)"
push_bit_of_0: 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 push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n zero = zero"
oproj_Field2: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "oproj r s f"   and "b \<in> Field s" shows "\<exists>a\<in>Field r. f a = b"
snd_lens_plus: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e" assumes "wb_lens x"   and "x \<bowtie> y" shows "snd\<^sub>L ;\<^sub>L (x +\<^sub>L y) = y"
e_psp_stable: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "F \<in> stable B"   and "\<forall>C\<in>CC. C \<inter> B \<in> CC" shows "F \<in> (A \<inter> B) leadsTo[CC] (A' \<inter> B)"
dist_metric_completion_def: shows "dist \<equiv> map_fun rep_metric_completion (map_fun rep_metric_completion id) (\<lambda>x y. lim (\<lambda>n. dist (x n) (y n)))"
lim3: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. (f n + g n) $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub (\<lambda>n. f n + g n) $$ (i, j)"
simps_2: fixes k :: "'a"   and v :: "'d"   and xs :: "('a \<times> 'd) list"   and x :: "'a" shows "lookup_pair ((k, v) # xs) x = (case comp x k of Eq \<Rightarrow> v | Lt \<Rightarrow> 0::'d | Gt \<Rightarrow> lookup_pair xs x)"
simple_closed_path_winding_number_inside: fixes \<gamma> :: "real \<Rightarrow> complex" assumes "simple_path \<gamma>" obtains "\<And>z. z \<in> inside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1"   | "\<And>z. z \<in> inside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = - 1"
fun_upd_None_restrict: fixes m :: "'a \<Rightarrow> 'b option"   and D :: "'a set"   and x :: "'a" shows "(m |` D)(x := None) = (if x \<in> D then m |` (D - {x}) else m |` D)"
assoc_naturality: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "arr f0"   and "arr f1"   and "arr f2" shows "\<a>[cod f0, cod f1, cod f2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) = (f0 \<otimes> f1 \<otimes> f2) \<cdot> \<a>[local.dom f0, local.dom f1, local.dom f2]"
YU: shows "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))))))) = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))) \<times> set (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))))"
inv_into_funcset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A = B" shows "restrict (inv_into A f) B \<in> B \<rightarrow> A"