hcju/isabelleps huggingface dataset:
append_delete_ids_imply_messages_same: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and ir :: "'a"   and e2 :: "'b" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Append i e1) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
eval_final: fixes 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 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 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)" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "final e'"
bij_imp_Collect_image: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "bij f" shows "{x. P (f x)} = inv f ` {y. P y}"
P_def: shows "P = (\<lambda>(k, ms). K k * (\<Prod>i<n. M (ms ! i)))"
otherassms_2: shows "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
itrm_trans: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and y :: "'a itrm"   and z :: "'a itrm" assumes "itrm_cong R x y"   and "itrm_cong R y z" shows "itrm_cong R x z"
and_mask: fixes w :: "'a word"   and n :: "nat" shows "w && mask n = w << size w - n >> size w - n"
collapse: fixes list :: "'a list" assumes "list \<noteq> []" shows "hd list # tl list = list"
that: fixes Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<parallel> P \<longmapsto> \<tau> \<prec> Q"   and "P' \<sim> Q \<parallel> !P" shows "thesis"
that_2: shows "g \<in> X \<rightarrow> topspace Z"
simps_9: shows "OUT sA9 = {data22, data23, data24}"
D2: shows "induced_function A f = (\<lambda>x. induced_function A g x - induced_function A h x)"
free_tm_def: fixes n :: "name" shows "free n = Hd (hd.Var (of_name n))"
eq_set_inc: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "a \<in> A"   and "A = B" shows "a \<in> B"
fmapU_WriterT: fixes f :: "udom \<rightarrow> udom"   and m :: "udom\<cdot>'f writer\<cdot>'e" shows "fmapU\<cdot>f\<cdot>(WriterT\<cdot>m) = WriterT\<cdot>(fmap\<cdot>(fmap\<cdot>f)\<cdot>m)"
SPRSingleAutoDFS_def: shows "SPRSingleAutoDFS \<equiv> \<lambda>kbp envInit envAction envTrans envVal envObs a. alg_dfs trie_odlist_MapOps trans_MapOps (SPRViewSingle.spr_simObs envObs a) (SPRViewSingle.spr_simTrans kbp envAction envTrans envVal envObs a) (SPRViewSingle.spr_simAction kbp envVal a) (map (SPRViewSingle.spr_simInit envInit envObs a \<circ> envObs a) envInit)"
in_hhomE: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr \<mu>" and "D.obj a" and "D.obj b" and "src\<^sub>D \<mu> = a" and "trg\<^sub>D \<mu> = b"
Limit_vsingleton_in_VsetI: fixes \<alpha> :: "V"   and a :: "V" assumes "Limit \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "ZFC_in_HOL.set {a} \<in>\<^sub>\<circ> Vset \<alpha>"
substT_rawpsubstT_fresh_switch: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list"   and s :: "'trm"   and y :: "'var" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> atrm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "substT (rawpsubstT r txs) s y = rawpsubstT r ((s, y) # txs)"
fst_annotate_rlen: fixes l :: "'b list" shows "map fst (annotate_rlen l) = rev [0..<length l]"
weight_spmf_eq_0: fixes p :: "'c spmf" shows "(weight_spmf p = 0) = (p = return_pmf None)"
case_1: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<sigma> \<Rightarrow> 'a"   and x1 :: "\<omega>" shows "(case \<omega>\<upsilon> x1 of \<omega>\<upsilon> x \<Rightarrow> f1 x | \<sigma>\<upsilon> x \<Rightarrow> f2 x) = f1 x1"
swap_def: fixes x :: "'d \<times> 'c" shows "swap x \<equiv> (snd x, fst x)"
euler_incr_fas_nth_def: fixes X0 :: "floatarith list"   and h :: "floatarith"   and CX :: "floatarith list"   and i :: "nat" shows "euler_incr_fas_nth X0 h CX i = X0 ! i + h * ode_fa_nth CX i"
phl_cons2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and w :: "'b"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.pre_dioid plus times less_eq less"   and "less_eq w x"   and "less_eq (times z y) (times y w)" shows "less_eq (times z y) (times y x)"
simps_1: shows "compare\<cdot>[]\<cdot>[] = EQ"
insert_remove: fixes A :: "'d set"   and x :: "'d" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
has_domain_iff_arr: fixes f :: "'a" shows "(V.domains f \<noteq> {}) = V.arr f"
fpxs_compose_power_eq_iff: fixes r :: "rat"   and f :: "'b fpxs"   and g :: "'b fpxs" assumes "0 < r" shows "(fpxs_compose_power f r = fpxs_compose_power g r) = (f = g)"
exec_c: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f"
ideal_prod_eq_inter: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R"   and "I <+> J = carrier R" shows "I \<cdot> J = I \<inter> J"
has_disc_alist_and: fixes disc :: "'b \<Rightarrow> bool"   and as :: "'b negation_type list" shows "has_disc disc (alist_and as) = (\<exists>a\<in>set as. has_disc disc (negation_type_to_match_expr a))"
is_ub_upward: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "po.is_ub below S x"   and "below x y" shows "po.is_ub below S y"
length_remdups_leq: fixes xs :: "'g list" shows "length (remdups xs) \<le> length xs"
in_set_least_squares_approximation_eq: fixes x :: "(real, 'a) vec"   and A :: "((real, 'a) vec, 'b) vec"   and b :: "(real, 'b) vec" shows "(x \<in> set_least_squares_approximation A b) = (Finite_Cartesian_Product.transpose A ** A *v x = Finite_Cartesian_Product.transpose A *v b)"
set_disconnected_nodes_ptr_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "h \<turnstile> ok set_disconnected_nodes document_ptr disc_nodes" shows "document_ptr |\<in>| document_ptr_kinds h"
QuadRes_set_prime: fixes p :: "int" assumes "prime p"   and "odd p" shows "{x. QuadRes p x \<and> x \<in> {0..<p}} = {x\<^sup>2 mod p |x. x \<in> {0..(p - 1) div 2}}"
bcomp_exits: fixes i :: "int"   and b :: "bexp"   and f :: "bool" assumes "0 \<le> i" shows "exits (bcomp b f i) \<subseteq> {size (bcomp b f i), i + size (bcomp b f i)}"
sigma_finite: fixes M :: "'a \<Rightarrow> 'b measure"   and I :: "'a set" assumes "product_sigma_finite M"   and "finite I" shows "sigma_finite_measure (Pi\<^sub>M I M)"
right_diff_distrib: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.ring plus zero minus uminus times" shows "times a (minus b c) = minus (times a b) (times a c)"
compare_double_le_0: fixes x :: "double"   and y :: "double" shows "(compare_double x y \<le> 0) = (Code_Double.is_nan x \<or> fcompare_double x y \<in> {ccode.Eq, ccode.Lt})"
"1_2": shows "e1 # e2 # es = map (\<lambda>i. (x i, x (i + j), local.neville_aitken_main i (i + j))) [l..<Suc (l + k)]"
pref_profile_unique_favorites_axioms: shows "pref_profile_unique_favorites agents alts R"
real_affinity_eq: fixes m :: "'b"   and x :: "'b"   and c :: "'b"   and y :: "'b" assumes "m \<noteq> (0::'b)" shows "(m * x + c = y) = (x = inverse m * y + - (c / m))"
Mg: shows "g \<in> stock_measure t' \<rightarrow>\<^sub>M stock_measure t''"
terminals_singleton: fixes u :: "symbol \<times> character list" shows "terminals [u] = [terminal_of_token u]"
dur_ev_exceedsI: fixes d :: "real"   and xs :: "('c \<Rightarrow> real) stream"   and t :: "real" assumes "\<forall>i. \<exists>j\<ge>i. d \<le> dur xs j - dur xs i"   and "0 < d" obtains i :: "nat" where "t < dur xs i"
extends_to_elementary_cartesian_category: shows "elementary_cartesian_category (\<cdot>) pr0 pr1 \<one> trm"
sets_lim: shows "sets PF.lim = sets (Pi\<^sub>M UNIV M)"
\<psi>_def: fixes ba :: "'c \<times> 'c" shows "\<psi>C ba = inv_into (C.hom (fst ba) (snd ba)) (\<phi>C ba)"
matrix_mul_rid: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "A ** mat (1::'d) = A"
construct_NofI_def: fixes p :: "real poly"   and I :: "real poly list" shows "construct_NofI p I = rat_of_int (changes_R_smods p (pderiv p * prod_list I))"
row_to_poly_mapping_of: fixes ts :: "('d, 'e) pp list"   and r :: "'f vec" assumes "distinct ts"   and "dim_vec r = length ts" shows "row_to_poly (map pp.mapping_of ts) r = Poly_Mapping.map_key PP (row_to_poly ts r)"
id3_2: shows "k - (k - i - 1) = Suc i"
approx_tse_form'_less: fixes prec :: "nat"   and t :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and s :: "nat"   and ivl :: "float interval"   and x :: "real" assumes "approx_tse_form' prec t (Add a (Minus b)) s ivl (\<lambda>ivl. 0 < lower ivl)"   and "x \<in>\<^sub>r ivl" shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
measurable_space: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and A :: "'d measure"   and x :: "'c" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "x \<in> space M" shows "f x \<in> space A"
remdups_map_remdups: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "remdups (list.map f (remdups xs)) = remdups (list.map f xs)"
bij_imp_bij_inv: fixes f :: "'d \<Rightarrow> 'e" assumes "bij f" shows "bij (inv f)"
sets_eq: shows "E.events = Pow (Pow E.S_edges)"
set_iterator_abs_trivial: shows "set_iterator_abs id (\<lambda>_. True) = set_iterator"
simps: fixes p :: "perm"   and x :: "'b"   and y :: "'c" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
append_subset_2: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "set xs = set (ys @ zs)" shows "set zs \<subseteq> set xs"
adjoint_scale: fixes a :: "'c"   and A :: "'c mat" shows "adjoint (a \<cdot>\<^sub>m A) = conjugate a \<cdot>\<^sub>m adjoint A"
comp_arr_inv': fixes f :: "'a arrow_of_spans_data" assumes "V.iso f" shows "f \<bullet> V.inv f = V.cod f"
assms: shows "\<forall>x\<in>vars\<^sub>t\<^sub>s ts. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
Nats_cases: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b" assumes "class.semiring_1 one times plus zero"   and "x \<in> semiring_1.Nats one plus zero" obtains n :: "nat" where "x = semiring_1.of_nat one plus zero n"
equiv_false: assumes "\<not> ((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<or> (\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<or> (\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)))" shows "False"
sortedConcEnd: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "FWNormalisationCore.sorted (a # p) l"   and "in_list a l"   and "all_in_list p l"   and "singleCombinators [a]"   and "singleCombinators p" shows "FWNormalisationCore.sorted p l"
coreflexive_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x"   and "coreflexive y" shows "coreflexive (x \<squnion> y)"
int_simps_32: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" shows "\<turnstile> (\<not> Q \<longrightarrow> \<not> P) = (P \<longrightarrow> Q)"
wellformed_transaction_strand_unlabel_memberD_7: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step"   and t :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" assumes "wellformed_transaction T"   and "x \<in> set (unlabel (transaction_strand T))"   and "x = delete\<langle>t,s\<rangle>" shows "x \<in> set (unlabel (transaction_updates T))"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "page_t \<Rightarrow> 'a" shows "(case PREP of PREP \<Rightarrow> f1 | WAIT \<Rightarrow> f2 | BUF x \<Rightarrow> f3 x) = f1"
code: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat option"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and i :: "(String.literal \<times> value list) stream" shows "make_full_observation e s d p i = (let (s', o', d') = ltl_step e s d (shd i) in \<lparr>statename = s, datastate = d, action = shd i, output = p\<rparr> ## make_full_observation e s' d' o' (stl i))"
tail_invar_cons_up: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list"   and t' :: "('c, 'd) SkewBinomialTree" assumes "tail_invar (t # bq)"   and "rank t' < rank t"   and "tree_invar t'" shows "tail_invar (t' # t # bq)"
prime_elem_gauss_int_of_nat: fixes n :: "nat" assumes "prime n"   and "[n = 3] (mod 4)" shows "prime_elem (of_nat n)"
antisym_conv3: fixes x :: "'a list"   and y :: "'a list" assumes "\<not> x <lex y" shows "(\<not> y <lex x) = (x = y)"
eval_Pr_converg_Suc: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs"   and "eval (Pr n f g) (x # xs) \<down>" shows "eval (Pr n f g) (Suc x # xs) = eval g (x # the (eval (Pr n f g) (x # xs)) # xs)"
subterm_reduction_applicable_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set"   and t :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.subterm_reduction_applicable trm_ord sel S t = (is_compound t \<and> (basic_superposition.trm_rep trm_ord sel (lhs t) S \<noteq> lhs t \<or> basic_superposition.trm_rep trm_ord sel (rhs t) S \<noteq> rhs t))"
simps: fixes t :: "nat"   and a1 :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)" shows "clstep t a1 a2 = (\<exists>c n ts c' n' s'. a1 = (c, n, ts) \<and> a2 = (c', n', ts(t := s')) \<and> t < N \<and> cstep_sng (c, n, ts t) (c', n', s'))"
assms_4: shows "q \<equiv> Poly (map (\<lambda>k. \<Sum>i\<le>min N k. c (N - i) * f (k - i)) [0..<N + m])"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> P" shows "Block a = A"
par_strict_neq2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "C \<noteq> D"
Abs_eq_apps_conv: fixes x :: "name"   and r :: "lam"   and s :: "lam"   and ss :: "lam list" shows "((Lam [x].r) = s \<degree>\<degree> ss) = ((Lam [x].r) = s \<and> ss = [])"
simps_36: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stmt.If x51 x52 x53"
min_elems1: fixes s :: "'a event list"   and P :: "'a process"   and c :: "'a event" assumes "s \<notin> D P"   and "s @ [c] \<in> D P" shows "s @ [c] \<in> min_elems (D P)"
set_eq: shows "set (find_consistent_signs qs) = {map (\<lambda>x. if poly x 0 < 0 then - 1 else if poly x 0 = 0 then 0 else 1) qs}"
closed_map_snd: fixes S :: "'a set"   and T :: "'b set"   and c :: "('a \<times> 'b) set" assumes "compact S"   and "closedin (top_of_set (S \<times> T)) c" shows "closedin (top_of_set T) (snd ` c)"
encode_negative_transition_frame_axiom_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat"   and v :: "'a" shows "encode_negative_transition_frame_axiom \<Pi> t v \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O>; deleting_operators = filter (\<lambda>op. ListMem v (delete_effects_of op)) ops in \<^bold>\<not> (Atom (State t (index vs v))) \<^bold>\<or> (Atom (State (Suc t) (index vs v)) \<^bold>\<or> \<^bold>\<Or>map (\<lambda>op. Atom (Operator t (index ops op))) deleting_operators)"
flip_at_base_simps_1: fixes a :: "'a"   and b :: "'a" assumes "sort_of (atom a) = sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> a = b"
osafe_Sing: fixes \<theta> :: "('a, 'b) trm"   and x :: "'b" assumes "dfree \<theta>" shows "osafe (OSing x \<theta>)"
fps_divide_unit: fixes g :: "'a fps"   and f :: "'a fps" assumes "g $ 0 \<noteq> (0::'a)" shows "f / g = f * inverse g"
det_def: fixes f :: "'b \<Rightarrow> ('c \<times> 'b) \<times> bool" shows "det f \<equiv> \<forall>s. \<exists>r. f s = (r, False)"
dtabbrev: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and locs :: "locn \<times> locn"   and tvs :: "char list list"   and tn :: "char list"   and t0 :: "t" shows "evaluate_dec ck mn env s (Dtabbrev locs tvs tn t0) (s, Rval (make_sem_env nsEmpty nsEmpty))"
fvi_plus_bound: fixes b :: "nat"   and c :: "nat"   and \<phi> :: "formula"   and n :: "nat" assumes "\<forall>i\<in>fvi (b + c) \<phi>. i < n" shows "\<forall>i\<in>fvi b \<phi>. i < c + n"
subgraph_preserves_hom: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'c) labeled_graph"   and X :: "('b, 'd) labeled_graph"   and h :: "('d \<times> 'c) set" assumes "subgraph A B"   and "graph_homomorphism X A h" shows "graph_homomorphism X B h"
ExecCondTrue: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and e\<^sub>1 :: "expr"   and u\<^sub>1 :: "nat"   and e\<^sub>2 :: "expr"   and u\<^sub>2 :: "nat"   and s\<^sub>1 :: "stmt"   and \<rho> :: "(char list \<Rightarrow> nat option) option"   and s\<^sub>2 :: "stmt" assumes "eval \<sigma> e\<^sub>1 = Some u\<^sub>1"   and "eval \<sigma> e\<^sub>2 = Some u\<^sub>2"   and "u\<^sub>1 = u\<^sub>2"   and "s\<^sub>1 \<rhd> \<sigma> \<leadsto> \<rho>" shows "IfEq e\<^sub>1 e\<^sub>2 s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> \<rho>"
set_intros_2: fixes y :: "'c"   and x22 :: "'c list"   and x21 :: "'c" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
intro: fixes T :: "'a \<Rightarrow> 'b"   and V :: "'a set"   and W :: "'b set" assumes "bij_betw T V W" shows "VectorSpaceIso_axioms V T W"
cases: fixes x :: "nat \<times> cell list \<times> cell list" obtains st :: "nat" and l :: "cell list" and r :: "cell list" where "x = (st, l, r)"
bibd_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int" shows "bibd point_set block_collection u_block_size index \<equiv> t_design point_set block_collection u_block_size 2 index"
b_e_type_select: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Select" shows "\<exists>ts'' t. ts = ts'' @ [t, t, T_i32] \<and> ts' = ts'' @ [t]"
sigma_sets_mono'': fixes A :: "'c set"   and C :: "'c set"   and D :: "'c set set"   and B :: "'c set set" assumes "A \<in> sigma_sets C D"   and "B \<subseteq> D"   and "D \<subseteq> Pow C" shows "sigma_sets A B \<subseteq> sigma_sets C D"
sscan_shift: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c stream"   and a :: "'b" shows "sscan f (xs @- ys) a = scan f xs a @- sscan f ys (fold f xs a)"
F2_def: fixes ds :: "'d set"   and X' :: "'x set" shows "F2 ds X' = - Rf (CD_on ds) X'"
fls_divide_shift_numer_nonzero: fixes f :: "'b fls"   and m :: "int"   and g :: "'b fls" assumes "f \<noteq> 0" shows "fls_shift m f / g = fls_shift m (f / g)"
ee': shows "Inum (a # bs) e = Inum (a # bs) e'"
interleave_comm: fixes a :: "'b list"   and b :: "'b list" shows "a \<otimes> b = b \<otimes> a"
gauss_jordan_single_2: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C" shows "C \<in> carrier_mat nr nc"
bound_upD: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> up R" shows "\<exists>n. bound \<zero> n f"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "S.ide (g \<cdot>\<^sub>S f)" obtains "S.seq g f" and "S.seq f g" and "g \<cdot>\<^sub>S f = S.dom f" and "g \<cdot>\<^sub>S f = S.cod g"
abs_delete: fixes t :: "'b trie_map"   and xs :: "'b list" assumes "invar t" shows "Trie_Map.abs (Trie_Map.delete xs t) = Trie_Fun.delete xs (Trie_Map.abs t)"
to_tree_order_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and node :: "(_) object_ptr"   and nodes' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "node \<in> set nodes"   and "h \<turnstile> to_tree_order node \<rightarrow>\<^sub>r nodes'"   and "known_ptrs h"   and "type_wf h" shows "set nodes' \<subseteq> set nodes"
border_length_r_less: fixes ys :: "'b list" shows "\<forall>xs. strict_border xs ys \<longrightarrow> length xs < length ys"
quadratic_shape1b: fixes a :: "real"   and x :: "real"   and y :: "real"   and b :: "real"   and c :: "real"   and z :: "real" assumes "0 < a"   and "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"   and "y < z" shows "0 < a * z\<^sup>2 + b * z + c"
basis_wf_many: fixes b :: "real \<Rightarrow> real"   and b' :: "real \<Rightarrow> real"   and bs :: "(real \<Rightarrow> real) list" shows "basis_wf (b # b' # bs) = (filterlim b at_top at_top \<and> (\<lambda>x. ln (b' x)) \<in> o(\<lambda>x. ln (b x)) \<and> basis_wf (b' # bs))"
small_step_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> int option"   and x2a :: "com"   and x2b :: "char list \<Rightarrow> int option"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int option) \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> int option) \<Rightarrow> bool" assumes "(x1a, x1b) \<rightarrow> (x2a, x2b)"   and "\<And>a s i x. Def_Init_Exp.aval a s = Some i \<Longrightarrow> P (x ::= a) s SKIP (s(x \<mapsto> i))"   and "\<And>c s. P (SKIP;; c) s c s"   and "\<And>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. \<lbrakk>(c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s'); P c\<^sub>1 s c\<^sub>1' s'\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s (c\<^sub>1';; c\<^sub>2) s'"   and "\<And>b s c\<^sub>1 c\<^sub>2. Def_Init_Exp.bval b s = Some True \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s c\<^sub>1 s"   and "\<And>b s c\<^sub>1 c\<^sub>2. Def_Init_Exp.bval b s = Some False \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s c\<^sub>2 s"   and "\<And>b c s. P (WHILE b DO c) s (IF b THEN c;; WHILE b DO c ELSE SKIP) s" shows "P x1a x1b x2a x2b"
IH2: fixes m :: "'a simple_match" assumes "p \<notin> Collect (simple_matches m)" shows "simple_fw (rmshadow' rs (P \<union> Collect (simple_matches m))) p = simple_fw rs p"
partial_fun_upd: fixes f :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and y :: "'c" shows "(f(x \<mapsto> y)) x = Some y"