hcju/isabelleps huggingface dataset:
RedCondT: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>if (true) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e\<^sub>1,s,b\<rangle>"
zeros_f_def: shows "zeros_f \<equiv> {p. f p = 0}"
assms: shows "x \<rhd> y = y \<rhd> x"
intros_1: fixes x :: "real" assumes "x \<in> \<rat>" shows "x \<in> radical_sqrt"
strict_mono_less_sets_Min: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "strict_mono_sets I f"   and "finite I"   and "I \<noteq> {}" shows "f (Min I) \<lless> \<Union> (f ` (I - {Min I}))"
singular_relboundary_imp_chain: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S c" shows "singular_chain p X c"
swapEnvIm_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "swapEnvIm xs x y rho \<equiv> \<lambda>zs. lift (swap xs x y) (rho zs)"
compute_bound_fa_correct: fixes prec :: "nat"   and f :: "floatarith"   and I :: "float interval list"   and ivl :: "float interval"   and i :: "real list" assumes "compute_bound_fa prec f I = Some ivl"   and "i all_in I" shows "interpret_floatarith f i \<in>\<^sub>r ivl"
Cons_2: fixes i' :: "nat"   and j' :: "nat"   and k :: "nat" assumes "i' < length (y # xs)"   and "j' < length (y # xs)"   and "(y # xs) ! i' = Some k" shows "((y # xs) ! j' = Some k) = (\<not> tr_lookup T (i' + jj) (j' + jj))"
eq_\<I>_restrict_gpv_full: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I> gpv gpv'" shows "eq_\<I>_gpv A \<I>_full (restrict_gpv \<I> gpv) (restrict_gpv \<I> gpv')"
\<psi>_char_3: shows "F'.\<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = F'.\<iota>\<^sub>1 \<cdot>\<^sub>D D'.T (F'.\<psi>, F'.\<psi>)"
p: shows "prv (imp (PP \<langle>\<phi>\<rangle>) \<phi>)"
IH: fixes ns' :: "'node list" assumes "suffix ns' ns"   and "ns' \<noteq> []" shows "EntryPath g ns'"
irreducible\<^sub>d_dvd_smult: fixes p :: "'c poly"   and q :: "'c poly" assumes "0 < degree p"   and "irreducible\<^sub>d q"   and "p dvd q" shows "\<exists>c. c \<noteq> (0::'c) \<and> q = Polynomial.smult c p"
convergent_Cauchy: fixes "open" :: "'e set \<Rightarrow> bool"   and uniformity :: "('e \<times> 'e) filter"   and X :: "nat \<Rightarrow> 'e" assumes "class.uniform_space open uniformity"   and "topological_space.convergent open X" shows "uniform_space.Cauchy uniformity X"
swaprows_mat_def: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "swaprows_mat n k l = mat n n (\<lambda>(i, j). if k = i \<and> l = j \<or> k = j \<and> l = i \<or> i = j \<and> i \<noteq> k \<and> i \<noteq> l then 1::'b else (0::'b))"
that: fixes W :: "'a set" assumes "W \<in> I"   and "\<forall>U\<in>Us. W \<subseteq> U" shows "thesis"
replace_coeff_add: fixes f :: "'c \<Rightarrow> 'b"   and p1 :: "'c mpoly"   and p2 :: "'c mpoly" assumes "f (0::'c) = (0::'b)"   and "\<And>a b. f (a + b) = f a + f b" shows "replace_coeff f (p1 + p2) = replace_coeff f p1 + replace_coeff f p2"
af_simps_2: fixes w :: "'a set list" shows "af false w = false"
pointermap_get_validI: fixes m :: "('a, 'b) pointermap_scheme"   and a :: "'a"   and p :: "nat" assumes "pointermap_sane m"   and "getentry m a = Some p" shows "pointermap_p_valid p m"
expand: fixes stream :: "'d stream"   and stream' :: "'d stream" assumes "shd stream = shd stream' \<and> stl stream = stl stream'" shows "stream = stream'"
map_prod_def: fixes f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
asIMOD_igSwapIGAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapIGAbs (asIMOD SEM)"
ordertype_UU_ge_\<omega>2: shows "ordertype UNIV pair_less \<le> ordertype UU pair_less"
R2_S2_False: fixes \<sigma> :: "bool"   and b0 :: "bool"   and b1 :: "bool"   and D2 :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf" assumes "(if \<sigma> then b0 else b1) = False" shows "spmf (etp.R2 (b0, b1) \<sigma> \<bind> D2) True = spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). etp.S2 \<sigma> out2 \<bind> D2)) True"
inner_diff_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (minus x y) z = inner x z - inner y z"
appNone: fixes i :: "instr"   and P :: "'b prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and et :: "ex_table" shows "app i P mxs T\<^sub>r pc mpc et None = True"
Un_mono: fixes A :: "'c set"   and C :: "'c set"   and B :: "'c set"   and D :: "'c set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
transitive_closure_trans_7: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
hyps_3: assumes "P \<turnstile> D \<preceq>\<^sup>* E" shows "P \<turnstile> D has F:T (fm) in E"
ordIso_iff_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
rbt_lookup_rbt_map_entry: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (ord.rbt_map_entry less k f t) = (ord.rbt_lookup less t)(k := map_option f (ord.rbt_lookup less t k))"
iterates_def: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" shows "iterates body G i = ((\<lambda>x. wp (body ;; Embed x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) ^^ i) (\<lambda>P s. 0)"
BoxPrime: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> \<box>P \<longrightarrow> \<box>($P \<and> P$)"
Tag_Sign: fixes X :: "tag"   and X' :: "msg"   and Y' :: "msg" shows "Tag X \<noteq> Sign X' Y'"
fmeasurable_def: fixes M :: "'b measure" shows "fmeasurable M = {A \<in> sets M. emeasure M A < \<infinity>}"
assms: fixes c :: "'a le_constraint" assumes "c \<in> set cs" shows "v \<Turnstile>\<^sub>l\<^sub>e c"
remove1_retains_prod: fixes a :: "'a"   and xs :: "'a list" shows "a \<in> set xs \<longrightarrow> prod_list xs = prod_list (remove1 a xs) * a"
prems_2: shows "(a, b) = chophelper (t # L) L'"
\<sigma>_bij_dom_img: shows "bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>)"
DynPDG_ddep_edge_CFG_path_1: fixes n :: "'node"   and V :: "'var"   and as :: "'edge list"   and n' :: "'node" assumes "n -{V}as\<rightarrow>\<^sub>d\<^sub>d n'" shows "n -as\<rightarrow>* n'"
is_nth_power_def: fixes n :: "nat"   and x :: "'b" shows "is_nth_power n x = (\<exists>y. x = y ^ n)"
length_removeAll_less: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "length (removeAll x xs) < length xs"
arr_Par_comp_Par: fixes \<alpha> :: "V"   and S :: "V"   and T :: "V" assumes "arr_Par \<alpha> S"   and "arr_Par \<alpha> T" shows "arr_Par \<alpha> (S \<circ>\<^sub>R\<^sub>e\<^sub>l T)"
mset_subset_eq_count: fixes A :: "'c multiset"   and B :: "'c multiset"   and a :: "'c" assumes "A \<subseteq># B" shows "count A a \<le> count B a"
NewHeapElem_typed: fixes t :: "'thread_id"   and x :: "'x"   and h :: "'heap"   and ta :: "('addr, 'thread_id, 'x, 'heap) Jinja_thread_action"   and x' :: "'x"   and h' :: "'heap"   and ad :: "'addr"   and CTn :: "htype" assumes "t \<turnstile> \<langle>x, h\<rangle> -ta\<rightarrow> \<langle>x', h'\<rangle>"   and "NewHeapElem ad CTn \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"   and "typeof_addr ad \<noteq> None" shows "typeof_addr ad = \<lfloor>CTn\<rfloor>"
set_map: fixes f :: "'f \<Rightarrow> 'd"   and v :: "'f set!['e]" shows "set_nebset (map_nebset f v) = f ` set_nebset v"
suc_def: fixes A :: "'b set" shows "s.suc A \<equiv> s.minim (s.AboveS A)"
simps_5: fixes a :: "MIR.num" shows "bound0 (Gt a) = numbound0 a"
measurable_Jordan: fixes S :: "'c set" assumes "bounded S"   and "negligible (frontier S)" shows "S \<in> lmeasurable"
map_pair_subset: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "'b \<times> 'd \<Rightarrow> 'b \<times> 'c"   and xs :: "('b \<times> 'd) list" assumes "comparator comp" shows "set (map_pair f xs) \<subseteq> f ` set xs"
igSwapIGOpSTR_def: fixes MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'l, 'm) model_scheme" shows "igSwapIGOpSTR MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
feasible_unitaryD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "feasible t"   and "unitary P" shows "unitary (t P)"
assms: fixes C :: "'a set" assumes "chain_on (\<sqsubset>) C A"   and "C \<noteq> {}" shows "\<exists>x\<in>A. glb (\<sqsubset>) C x"
s1_2: shows "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
aHom_def: fixes A :: "('e, 'g) aGroup_scheme"   and B :: "('f, 'h) aGroup_scheme" shows "aHom A B = {f \<in> carrier A \<rightarrow> carrier B. f \<in> extensional (carrier A) \<and> (\<forall>a\<in>carrier A. \<forall>b\<in>carrier A. f (a \<plusminus>\<^bsub>A\<^esub> b) = f a \<plusminus>\<^bsub>B\<^esub> f b)}"
ee_rule: assumes "0 \<in> {0..1}"   and "0 \<in> {0..1}"   and "0 \<in> {0..1}"   and "\<bar>0 - 0\<bar> < ee 0"   and "\<bar>0 - 0\<bar> < ee 0" shows "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and> (\<forall>u\<in>{0..1}. cmod (g1 u - k (0, u)) < d \<and> cmod (g2 u - k (0, u)) < d) \<and> linked_paths atends g1 g2 \<longrightarrow> contour_integral g2 f = contour_integral g1 f"
distinct: shows "distinct [0\<^sub>h, of_complex (cor x), of_complex (cor y), inversion (of_complex (cor x))]"
row_rank_def_gen: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A \<equiv> vec.dim (rows A)"
simps: fixes a1 :: "'a trm"   and a2 :: "'a trm" shows "a1 >> a2 = ((\<exists>A. a1 = A \<and> a2 = A) \<or> (\<exists>A A' B B' x T. a1 = App (Fn x T A) B \<and> a2 = A'[x ::= B'] \<and> A >> A' \<and> B >> B') \<or> (\<exists>A A' x T. a1 = Fn x T A \<and> a2 = Fn x T A' \<and> A >> A') \<or> (\<exists>A A' B B'. a1 = App A B \<and> a2 = App A' B' \<and> A >> A' \<and> B >> B') \<or> (\<exists>A A' B B'. a1 = SimplyTyped.Pair A B \<and> a2 = SimplyTyped.Pair A' B' \<and> A >> A' \<and> B >> B') \<or> (\<exists>P P'. a1 = Fst P \<and> a2 = Fst P' \<and> P >> P') \<or> (\<exists>A A' B. a1 = Fst (SimplyTyped.Pair A B) \<and> a2 = A' \<and> A >> A') \<or> (\<exists>P P'. a1 = Snd P \<and> a2 = Snd P' \<and> P >> P') \<or> (\<exists>B B' A. a1 = Snd (SimplyTyped.Pair A B) \<and> a2 = B' \<and> B >> B'))"
rerr_invalid_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and m :: "msg" shows "rerr_invalid crt m \<equiv> case m of Rerr destsc x \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> iD crt \<and> the (destsc ripc) = sqn crt ripc | _ \<Rightarrow> True"
P1: shows "B Midpoint C C' \<and> Cong A C A C'"
limit_consume_function: fixes f :: "'a set \<Rightarrow> 'b set"   and P :: "'a set \<Rightarrow> bool"   and g :: "'a set \<Rightarrow> 'a set"   and h :: "'b set \<Rightarrow> 'b set"   and I :: "'a set" assumes "continuous f"   and "\<And>X. P X \<Longrightarrow> f (g X) = h (f X) \<and> P (g X)"   and "setmonotone g"   and "P I" shows "f (limit g I) = limit h (f I)"
add_mset_eq_singleton_iff: fixes x :: "'k"   and M :: "'k multiset"   and y :: "'k" shows "(add_mset x M = {#y#}) = (M = {#} \<and> x = y)"
simps_1: shows "polypow 0 = (\<lambda>p. (1::'b)\<^sub>p)"
set_3: fixes x1 :: "'h" shows "generat_outs (Pure x1) = {}"
bounded_closed_interval: fixes a :: "'c"   and b :: "'c" shows "bounded {a..b}"
nat_of_string_string_of_nat: fixes n :: "nat" shows "nat_of_string (string_of_nat n) = n"
fixp_induct_uc: fixes U :: "'c \<Rightarrow> 'a \<Rightarrow> 'b Heap"   and F :: "'c \<Rightarrow> 'c"   and C :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'c"   and f :: "'c"   and P :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> bool" assumes "\<And>x. mono_Heap (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (heap.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "heap.admissible P"   and "P (\<lambda>_. Heap Map.empty)"   and "\<And>f. P (U f) \<Longrightarrow> P (U (F f))" shows "P (U f)"
node_deliver_messages_append: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and xs :: "'c event list"   and ys :: "'c event list" assumes "network history msg_id" shows "trivial_network.node_deliver_messages (xs @ ys) = trivial_network.node_deliver_messages xs @ trivial_network.node_deliver_messages ys"
ffold_finsert: fixes x :: "'a"   and A :: "'a fset"   and z :: "'b" assumes "x |\<notin>| A" shows "ffold f z (finsert x A) = f x (ffold f z A)"
circlepath: fixes z :: "complex"   and r :: "real" shows "circlepath z r = (\<lambda>x. z + complex_of_real r * exp (2 * complex_of_real pi * \<i> * complex_of_real x))"
inv_cuts: fixes xs :: "int list"   and c :: "int list"   and cs :: "int list list" assumes "cuts xs = c # cs" shows "\<exists>ys. c = Challenge1A.take xs \<and> xs = c @ ys \<and> cs = cuts ys"
elims: fixes x :: "'v node \<times> 'v node"   and y :: "ennreal" assumes "capacity' x = y" obtains xa :: "'v" where "x = (SOURCE, Inner xa__)" and "y = (if xa__ = source \<Delta> then \<alpha> else 0)"   | xa :: "'v" and ya :: "'v" where "x = (Inner xa, Inner ya)" and "y = capacity \<Delta> (xa, ya)"   | vb :: "'v" where "x = (Inner vb, SOURCE)" and "y = 0"   | v :: "'v node" where "x = (v, SOURCE)" and "y = 0"
productive_imp_produces_Max_literal: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "ground_resolution_with_selection.production S N C = {atm_of (Max_mset C)}"
order_iff_strict: fixes a :: "'b list"   and b :: "'b list" shows "subseq a b = (strict_subseq a b \<or> a = b)"
15: shows "changes_R_smods p (pderiv p * q) = int (card {x. poly p x = 0 \<and> 0 < poly q x}) + - 1 * int (card {x. poly p x = 0 \<and> poly q x < 0})"
bernoulli_stream_element_prob_rec: fixes M :: "bool stream measure"   and p :: "real"   and n :: "nat"   and x :: "bool stream" assumes "M = bernoulli_stream p"   and "0 \<le> p"   and "p \<le> 1" shows "emeasure M {w \<in> space M. stake (Suc n) w = stake (Suc n) x} = emeasure M {w \<in> space M. stake n w = stake n x} * ennreal (prob_component p x n)"
trivial_same: fixes X :: "'a set" shows "trivial X = (\<forall>x1\<in>X. \<forall>x2\<in>X. x1 = x2)"
lselect_lselect': fixes t :: "nat set"   and s :: "nat set"   and w :: "'b llist" assumes "\<And>i. i \<in> t \<Longrightarrow> enat i < esize s" shows "lselect t (lselect s w) = lselect (nth_least s ` t) w"
put_put: fixes s :: "'b"   and s' :: "'b"   and m :: "'m" shows "put s (put s' m) = put s' m"
indexed_eval_aux_in_carrier: fixes Ps :: "('d multiset \<Rightarrow> 'a) list"   and i :: "'d" assumes "list_all carrier_coeff Ps" shows "carrier_coeff (indexed_eval_aux Ps i)"
singleton_nonzero: fixes p :: "'a" assumes "singleton p" shows "p \<noteq> 0"
contour_integral_local_primitive_lemma: fixes g :: "real \<Rightarrow> complex"   and a :: "real"   and b :: "real"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex" assumes "g piecewise_differentiable_on {a..b}"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_field_derivative f' x) (at x within S)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> S" shows "(\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b}"
bottom_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c" shows "bottom r B = (\<forall>x. B \<sqsubseteq>\<^bsub>r\<^esub> x)"
is_nextElem_edges_eq: fixes f :: "face"   and a :: "nat"   and b :: "nat" assumes "distinct (vertices f)" shows "((a, b) \<in> \<E> f) = is_nextElem (vertices f) a b"
iterateoi_def: fixes olist_it :: "'e \<Rightarrow> ('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list"   and S :: "'e" shows "poly_set_iterateoi_defs.iterateoi olist_it S \<equiv> it_to_it (olist_it S)"
boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
hyps: assumes "i \<le> j"   and "i \<le> j - k" shows "[i..<j] = [i..<j - k] @ [j - k..<j]"
poly_strict_mono_def: fixes p :: "('d monom \<times> 'a) list"   and v :: "'d" shows "poly_strict_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<alpha> v \<succ> \<beta> v \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<beta> p"
assms: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and s :: "'c"   and a :: "'d"   and \<sigma>' :: "'a \<Rightarrow> 'e"   and s' :: "'g"   and \<zeta> :: "'a \<Rightarrow> 'b" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> T"   and "\<forall>j\<in>J. \<zeta> j = \<sigma> j" shows "\<exists>\<zeta>'. (\<forall>j\<in>J. \<zeta>' j = \<sigma>' j) \<and> ((\<zeta>, s), a, \<zeta>', s') \<in> T"
aux1: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set"   and U :: "('c \<Rightarrow> real) set set" assumes "u \<in> R"   and "R \<in> \<R>\<^sub>\<beta>"   and "U \<subseteq> \<R>\<^sub>\<beta>"   and "u \<in> \<Union> U" shows "R \<subseteq> \<Union> U"
nf_inres_SPEC: fixes \<Phi> :: "'b \<Rightarrow> bool"   and x :: "'b" shows "nf_inres (SPEC \<Phi>) x = \<Phi> x"
HCollect_def: fixes P :: "hf \<Rightarrow> bool"   and A :: "hf" shows "HCollect P A = (THE z. \<forall>u. (u \<^bold>\<in> z) = (P u \<and> u \<^bold>\<in> A))"
term_variants_P: fixes T :: "('a, 'b) Term.term list"   and S :: "('a, 'b) Term.term list"   and P :: "'a \<Rightarrow> 'a list"   and g :: "'a"   and f :: "'a" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)"   and "g \<in> set (P f)" shows "term_variants_pred P (Fun f T) (Fun g S)"
sinvar_mono_def: shows "sinvar_mono = (\<forall>nP N E' E. wf_graph \<lparr>nodes = N, edges = E\<rparr> \<and> E' \<subseteq> E \<and> sinvar \<lparr>nodes = N, edges = E\<rparr> nP \<longrightarrow> sinvar \<lparr>nodes = N, edges = E'\<rparr> nP)"
con1: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and vs :: "v list"   and v1 :: "v"   and ck :: "bool"   and s :: "'b state"   and s' :: "'b state" assumes "do_con_check (c env) cn (length es) \<and> build_conv (c env) cn (rev vs) = Some v1 \<and> evaluate_list ck env s (rev es) (s', Rval vs)" shows "evaluate ck env s (Con cn es) (s', Rval v1)"
MM2_axioms_def: shows "MM2_axioms \<equiv> (\<forall>x y. PP x y = (P x y \<and> x \<noteq> y)) \<and> (\<forall>y x. PP y x \<longrightarrow> (\<exists>z. P z x \<and> \<not> O z y))"
ctor2_o_fold2: shows "ctor2 \<circ> fold2 (F1map id ctor1 ctor2) (F2map id ctor1 ctor2) = id"
mem_tangent_space: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "c_manifold charts k" shows "(X \<in> c_manifold.tangent_space charts k p) = (c_manifold.is_derivation charts k X p \<and> extensional0 (c_manifold.diff_fun_space charts k) X)"
p: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1" shows "p (\<lambda>x. h (t, x))"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 k, $C v, $Store t tp a off] : ts _> ts'"
IfF: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and pc :: "int"   and v :: "Val"   and NEXT :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and os :: "Val list"   and s :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int" assumes "get_ins M l = Some (iftrue pc)"   and "v \<noteq> TRUE"   and "NEXT = (os, s, h)"   and "ll = l + 1" shows "(M, l, (v # os, s, h), 1, ll, NEXT) \<in> Step"
freshen_def: fixes t :: "llam"   and p :: "name" shows "freshen t p \<equiv> vsub t 0 (lPar p)"
"4_2": fixes i :: "nat" assumes "i < length (x # xs)"   and "i < length (y # ys)"   and "fst (P ((x # xs) ! i) ((y # ys) ! i))" shows "fst (P' ((x # xs) ! i) ((y # ys) ! i))"
not_less_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
bot_is_action: shows "R (false \<turnstile> true) \<in> {p. is_CSP_process p}"
Cons_4: shows "sharing_consistent \<S> \<O> (x # sb)"
preserves_iso: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "D.iso (\<Phi>\<^sub>G f)"
assms_2: shows "\<forall>i j. Q i j \<longrightarrow> R i j"
enat_unfold_next: fixes stop :: "'a \<Rightarrow> bool"   and a :: "'a"   and "next" :: "'a \<Rightarrow> 'a" assumes "\<not> stop a" shows "enat_unfold stop next a = eSuc (enat_unfold stop next (next a))"
x: fixes y :: "complex"   and Y :: "complex_interval" assumes "y \<in>\<^sub>c Y" shows "x * y \<in>\<^sub>c X * Y"
bind_option_cong_code: fixes x :: "'m option"   and y :: "'m option"   and f :: "'m \<Rightarrow> 'n option" assumes "x = y" shows "Option.bind x f = Option.bind y f"
a1: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (write_state_reg_instr instr s1)) \<and> t2 = snd (fst (write_state_reg_instr instr s2))"
suc: fixes t :: "'trm" assumes "t \<in> trm" shows "suc t \<in> trm"
rank_invar_to_set: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" assumes "rank_invar (t # bq)" shows "\<forall>t'\<in>set bq. rank t < rank t'"
\<rho>: shows "\<guillemotleft>\<rho> : g \<Rightarrow> r \<star> f\<guillemotright>"
simps: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'a option"   and \<nu> :: "'b"   and m' :: "'c \<Rightarrow> 'a option" shows "\<downharpoonleft>\<^sub>x (m, \<nu>, m') = (the (m x), \<nu>, the (m' x))"
Smith_nx1_works: fixes A :: "'a mat"   and n :: "nat"   and P :: "'a mat"   and S :: "'a mat" assumes "A \<in> carrier_mat n 1"   and "(P, S) = Smith_nx1 A" shows "is_SNF A (P, S, 1\<^sub>m 1)"
simps_2: fixes v :: "'c"   and v' :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "in_tree v (T v' l r) = (v = v' \<or> in_tree v l \<or> in_tree v r)"
cfactor_correct: fixes l :: "'a list"   and P :: "pol"   and c :: "int" assumes "in_carrier l" shows "Ipol l P = Ipol l (fst (cfactor P c)) \<oplus> \<guillemotleft>c\<guillemotright> \<otimes> Ipol l (snd (cfactor P c))"
lengthOneImpliesOnlyElement: fixes l :: "'a list"   and a :: "'a" assumes "length l = 1"   and "a \<in> set l" shows "\<forall>a'. a' \<in> set l \<longrightarrow> a' = a"
simps_1: shows "normalized_nnf_match MatchAny = True"
frechet_derivative_const: fixes c :: "'b"   and a :: "'a" shows "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0::'b)"