hcju/isabelleps huggingface dataset:
sep_add_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj" shows "plus x zero = x"
statically_complete: fixes B :: "'f \<times> 'l"   and N :: "('f \<times> 'l) set" assumes "B \<in> Bot_FL"   and "std.saturated N"   and "entails_\<G> N {B}" shows "\<exists>B'\<in>Bot_FL. B' \<in> N"
distinct_1: shows "ConcatOp \<noteq> IndexOfOp"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs H" shows "rev_H a = a"
has_derivative_within_alt: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)))"
closure_sequential: fixes l :: "'b"   and S :: "'b set" shows "(l \<in> closure S) = (\<exists>x. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l)"
resolvent_is_resolvent_upon: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" assumes "resolvent P1 P2 C" shows "\<exists>A. C = resolvent_upon P1 P2 A"
coerce_sprod: shows "COERCE('e \<otimes> 'f, 'g \<otimes> 'h) = sprod_map\<cdot>COERCE('e, 'g)\<cdot>COERCE('f, 'h)"
divide_le_eq_numeral1_1: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / numeral w \<le> a) = (b \<le> a * numeral w)"
dec_i: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list" assumes "list_repr i ba xs"   and "i \<noteq> 0" shows "list_repr (i - 1) (dec_i ba) xs"
interpret_op_def: fixes oper :: "('f, 'e) operation"   and state :: "'e \<Rightarrow> 'f set \<times> 'f set" shows "\<langle>oper\<rangle> state \<equiv> let metadata = fst (state (op_elem oper)); files = snd (state (op_elem oper)); after = case oper of Create i e \<Rightarrow> (metadata \<union> {i}, files) | Delete is e \<Rightarrow> (metadata - is, files - is) | Append i e \<Rightarrow> (metadata, files \<union> {i}) | Expunge e mo i \<Rightarrow> (metadata \<union> {i}, files - {mo}) | Store e mo i \<Rightarrow> (metadata, insert i (files - {mo})) in Some (state(op_elem oper := after))"
list_nonMt_append: fixes s :: "'b list" assumes "s \<noteq> []" shows "\<exists>a t. s = t @ [a]"
IH: shows "filter (\<lambda>y. f y = k) (isort_key f xs) = filter (\<lambda>y. f y = k) xs"
homogeneousI: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>s t. \<lbrakk>s \<in> keys p; t \<in> keys p\<rbrakk> \<Longrightarrow> deg_pm s = deg_pm t" shows "homogeneous p"
corec_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> 'b llist"   and g222 :: "'a \<Rightarrow> 'a" assumes "p a" shows "corec_llist p g21 q22 g221 g222 a = LNil"
attach_callee_parallel_intercept: fixes callee1 :: "'a \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'a, 'h, 'j) gpv"   and callee2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('g \<times> 'b, 'i, 'k) gpv"   and oracle1 :: "'c \<Rightarrow> 'h \<Rightarrow> ('j \<times> 'c) spmf"   and oracle2 :: "'c \<Rightarrow> 'i \<Rightarrow> ('k \<times> 'c) spmf" shows "attach_callee (callee1 \<ddagger>\<^sub>I callee2) (oracle1 \<oplus>\<^sub>O oracle2) = map_fun rprodl (map_fun id (map_spmf (map_prod id lprodr))) (lift_state_oracle extend_state_oracle (attach_callee callee1 oracle1) \<oplus>\<^sub>O \<dagger>(attach_callee callee2 oracle2))"
exec_eval_mono: fixes s :: "state"   and c :: "stmt"   and n :: "nat"   and t :: "state"   and e :: "expr"   and v :: "val" shows "(s -c-n\<rightarrow> t \<longrightarrow> (\<forall>m\<ge>n. s -c-m\<rightarrow> t)) \<and> (s -e\<succ>v-n\<rightarrow> t \<longrightarrow> (\<forall>m\<ge>n. s -e\<succ>v-m\<rightarrow> t))"
n: shows "t' = t + real_of_int (2 * n) * pi"
sets_into_space: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and x :: "'c set" assumes "subset_class \<Omega> M"   and "x \<in> M" shows "x \<subseteq> \<Omega>"
MGU_is_mgu_singleton: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and u :: "('c, 'd) Term.term" shows "MGU \<theta> t u = is_mgu \<theta> {(t, u)}"
u_B_authenticates_A_r: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and B :: "agent"   and Ts :: "nat" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "B \<noteq> Tgs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Says A B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
Heap_axioms: shows "Heap empty is_empty of_list multiset as_tree remove_max"
iotahom: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<iota>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub> \<in> gHom (\<natural>\<^bsub>G\<^esub>H) (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N)"
monoid_def: fixes G :: "('e, 'f) monoid_scheme" shows "Group.monoid G \<equiv> ((\<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G) \<and> (\<forall>x y z. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> z \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z))) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x) \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x)"
2: shows "sup_monoid.sum (\<lambda>k. f (i, k)) {k. True} = f (i, i)"
assms: shows "((\<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst n_ns) \<union> new (fst n_ns) \<or> \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst n_ns)) \<and> (\<forall>nd\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns))) \<and> (\<forall>nd\<in>snd n_ns. \<forall>nd'\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<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)"
r_inv_ex: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<exists>y\<in>carrier P. x \<oplus>\<^bsub>P\<^esub> y = \<zero>\<^bsub>P\<^esub>"
hyps_3: shows "(r[x::=u] \<degree>\<degree> ss) \<degree> s \<rightarrow>\<^sub>s t \<degree> s'"
mktop_Lin: fixes L :: "(alt \<times> alt) set"   and x :: "alt" assumes "L \<in> Lin" shows "mktop L x \<in> Lin"
that: fixes x :: "'d"   and h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "thesis"
irrefl: fixes a :: "'a list" shows "\<not> a <lex a"
member_remove: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"
gen_cong_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "gen_cong R x y"   and "gen_cong R y z" shows "gen_cong R x z"
is_monomial_setD: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "is_monomial_set A"   and "p \<in> A" shows "is_monomial p"
subgroup_imp_finite_comm_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "finite_comm_group (G\<lparr>carrier := adjoin G H a, carrier := H\<rparr>)"
mono: fixes n :: "nat" shows "a (Suc n) \<le> a n"
drinks_def: shows "drinks \<equiv> {|((0, 1), select), ((1, 1), coin), ((1, 1), vend_fail), ((1, 2), vend)|}"
hyps: shows "card (listset XS) = prod_list (map card XS)"
of_complex_diff: fixes x :: "complex"   and y :: "complex" shows "of_complex (x - y) = of_complex x - of_complex y"
order_trans_rules_22: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x < y"   and "y \<le> z" shows "x < z"
field_op_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier K"   and "b \<in> carrier K"   and "b \<noteq> \<zero>" shows "-\<^sub>a a \<cdot>\<^sub>r b\<^bsup>\<hyphen> K\<^esup> = (-\<^sub>a a) \<cdot>\<^sub>r b\<^bsup>\<hyphen> K\<^esup>"
edge_to_def: fixes c :: "'form"   and "is" :: "nat list" shows "edge_to c is = (case rev is of [] \<Rightarrow> ((c, []), in_port_at (c, []) 0) | i # is \<Rightarrow> ((c, 0 # rev is), in_port_at (c, 0 # rev is) i))"
LeastI2_wellorder_ex: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
APP_def: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e" shows "f $ a \<equiv> f a"
diag_mat_mult_diag_mat: fixes B :: "'a mat"   and n :: "nat"   and D :: "'a mat" assumes "B \<in> carrier_mat n n"   and "D \<in> carrier_mat n n"   and "diagonal_mat B"   and "diagonal_mat D" shows "B * D = mat n n (\<lambda>(i, j). if i = j then B $$ (i, i) * D $$ (i, i) else (0::'a))"
Henstock_lemma_part2: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and e :: "real"   and d :: "'a \<Rightarrow> 'a set"   and p :: "('a \<times> 'a set) set" assumes "f integrable_on cbox a b"   and "0 < e"   and "gauge d"   and "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b; d fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e"   and "p tagged_partial_division_of cbox a b"   and "d fine p" shows "(\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) \<le> 2 * real DIM('b) * e"
permute_profile_def: fixes \<sigma> :: "'f \<Rightarrow> 'd"   and R :: "'c \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'e" shows "permute_profile \<sigma> R = (\<lambda>i x y. R i (inv \<sigma> x) (inv \<sigma> y))"
get_trms_def: fixes C :: "'a literal set"   and E :: "'a trm set"   and t :: "inferences" shows "get_trms C E t = (if t = FirstOrder then filter_trms C E else E)"
module_incl_imp_submodule: fixes H :: "'c set" assumes "H \<subseteq> carrier M"   and "module R (md H)" shows "Module.submodule H R M"
Osum_embed: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Well_order r"   and "Well_order r'" shows "embed r (r \<union>o r') id"
dim_row_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a vec" shows "dim_row (mat\<^sub>r nr nc g) = nr"
moebius_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "moebius a b c d \<equiv> \<lambda>z. (a * z + b) / (c * z + d)"
LIMSEQ_Suc: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "f \<longlonglongrightarrow> l" shows "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
simps_3: fixes v :: "'b local.term"   and va :: "'b local.term"   and \<mu> :: "'b" shows "(v \<^bold>\<star> va) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = v \<^bold>\<star> va"
in_hhom_def: fixes \<mu> :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" shows "in_hhom \<mu> a b \<equiv> arr \<mu> \<and> src \<mu> = a \<and> trg \<mu> = b"
\<beta>': shows "\<beta> = \<delta> @ \<beta>'"
sv01_3: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1"   and "n \<noteq> 0"   and "the (inconsist (amalg01 f) (f \<triangleright> n)) < Suc n"   and "pdec1 (the (parallel (the (f 0)) (the (f 1)) (the (inconsist (amalg01 f) (f \<triangleright> n))))) = 0" shows "s\<^bsub>01\<^esub> (f \<triangleright> n) = f 1"
fresh_2: fixes a :: "NominalSequents.var"   and x2 :: "form"   and x1 :: "form_list" shows "a \<sharp> FCons x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
rel_Grp: fixes A1 :: "'a set"   and f1 :: "'a \<Rightarrow> 'c"   and A2 :: "'b set"   and f2 :: "'b \<Rightarrow> 'd" shows "rel_prod (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> A1 \<and> Basic_BNFs.snds x \<subseteq> A2} (map_prod f1 f2)"
simCasesCont: fixes Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and P :: "pi"   and C :: "'a" assumes "eqvt Rel"   and "\<And>a x Q'. \<lbrakk>Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> a; x \<sharp> C\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>[Rel] Q"
singleton_singletonT: fixes p :: "'a" assumes "singleton p" shows "p ; p\<^sup>T \<le> 1'"
x: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d x"
64227: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - y \<sqinter> - - x) = - x"
f_2: fixes epsilon :: "real" assumes "0 < epsilon" shows "emeasure M (f epsilon) < ennreal epsilon"
Un_commute: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = B \<union> A"
pseudo_ordered_set_axioms: shows "pseudo_ordered_set X (\<sqsubseteq>)"
assms_3: fixes y :: "V" shows "ZFC_in_HOL.succ y \<noteq> i"
size_neq: fixes x :: "('a, 'b) phantom" shows "size x \<noteq> 0"
is_poincare_line_not_trough_zero_not_trough_infty: fixes l :: "circline" assumes "is_poincare_line l"   and "0\<^sub>h \<notin> circline_set l" shows "\<infinity>\<^sub>h \<notin> circline_set l"
mbinop_def: shows "mbinop = 1\<^sub>\<nat>"
pochhammer_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a 1 = a"
f_def: shows "f = (\<lambda>i. if i \<in> I then 1 else 0)"
sample_coprime_lem: fixes x :: "nat"   and y :: "nat" assumes "x \<in> set_spmf sample_primes"   and "y \<in> set_spmf (sample_set_excl prime_set {x})" shows "lossless_spmf (sample_coprime ((x - Suc 0) * (y - Suc 0)))"
le_eq_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq a (uminus b) = less_eq (plus a b) zero"
uni: shows "unifier\<^sub>l\<^sub>s \<sigma> L"
span_base: fixes a :: "'c"   and S :: "'c set" assumes "a \<in> S" shows "a \<in> ideal S"
lc_monomial: fixes t :: "'t"   and c :: "'d" shows "lc (monomial c t) = c"
rtrancl_into_trancl1: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
assms: shows "- (x \<squnion> y) = - y \<or> - (- (x \<squnion> - y) \<squnion> - y) = x"
induct: fixes P :: "('b, 'c) Combinators \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> P (x \<oplus> y)"   and "P DenyAll"   and "\<And>v va. P (DenyAllFromTo v va)"   and "\<And>v va vb. P (AllowPortFromTo v va vb)" shows "P a0"
ap_state_rev_pure_B: fixes f :: "'f \<Rightarrow> 'e"   and x :: "('d, 'f) state" shows "pure f \<diamondop> B x = B (pure f \<diamondop> x)"
length_take: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "length (take n xs) = n"
eq: fixes u :: "real" shows "(SOME t. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - u) *\<^sub>R c + u *\<^sub>R x) = u"
makespan_mono_2: fixes m :: "nat"   and T :: "nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "LoadBalancing m"   and "T x \<le> y" shows "f_Max\<^sub>0 T m \<le> f_Max\<^sub>0 (T(x := y)) m"
f_le_lfp: fixes i :: "nat" shows "(f ^^ i) bot \<le> lfp f"
iter_narrow_pfp: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and p0 :: "'a acom"   and p :: "'a acom" assumes "\<And>x1 x2. \<lbrakk>P x1; P x2; x1 \<le> x2\<rbrakk> \<Longrightarrow> f x1 \<le> f x2"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<triangle> x2)"   and "P p0"   and "f p0 \<le> p0"   and "iter_narrow f p0 = Some p" shows "P p \<and> f p \<le> p"
simps: fixes a1 :: "Lambda_Example.type list"   and a2 :: "dB"   and a3 :: "Lambda_Example.type" shows "(a1 \<turnstile> a2 : a3) = ((\<exists>env x T. a1 = env \<and> a2 = Var x \<and> a3 = T \<and> nth_el1 env x T) \<or> (\<exists>T env t U. a1 = env \<and> a2 = Abs T t \<and> a3 = T \<Rightarrow> U \<and> T # env \<turnstile> t : U) \<or> (\<exists>env s U T t. a1 = env \<and> a2 = s \<degree> t \<and> a3 = U \<and> env \<turnstile> s : U \<Rightarrow> T \<and> env \<turnstile> t : T))"
nths_vars1': shows "nths c.dims0 c.vars1' = c.dims1"
insert_partition: fixes x :: "'b set"   and F :: "'b set set" assumes "x \<notin> F"   and "\<forall>c1\<in>insert x F. \<forall>c2\<in>insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}" shows "x \<inter> \<Union> F = {}"
lnths_empty: fixes xs :: "'c llist" shows "lnths xs {} = LNil"
PiE_restrict: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'h \<Rightarrow> 'i set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
bounded_antilinear_from_conjugate_space: shows "bounded_antilinear from_conjugate_space"
u_def: fixes n :: "nat" shows "u n = dist (0::'a) ((T ^^ n) (0::'a))"
supp_3: fixes fm1 :: "fm"   and fm2 :: "fm" shows "supp (fm1 OR fm2) = supp fm1 \<union> supp fm2"
\<pi>_eq_0: fixes x :: "real" assumes "x < 2" shows "\<pi> x = 0"
Uniq_def: fixes P :: "'b \<Rightarrow> bool" shows "Uniq P \<equiv> \<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x"
while_def: fixes p :: "'b Assertion"   and x :: "'b" shows "(While p do x) = ([\<cdot> p ] * x) ^ \<omega> * [\<cdot> - p ]"
assms_2: shows "set_offending_flows G nP \<noteq> {}"
smult_in_span: fixes W :: "'a vec set"   and x :: "'a vec"   and c :: "'a" assumes "W \<subseteq> carrier_vec n"   and "x \<in> local.span W" shows "c \<cdot>\<^sub>v x \<in> local.span W"
a: shows "List.member (concat (map (\<lambda>x. map (map_fun Rep_fin_string Abs_fin_string (string_cons x)) (fin_string_upto n)) enum_class.enum)) L"
lex_square_def: shows "lex_square \<equiv> less_than <*lex*> lex_triple"
Beta_commute: fixes a :: "'a"   and b :: "'a" shows "Beta a b = Beta b a"
eq_\<I>_gpv_inline1: fixes s1 :: "'s1"   and s2 :: "'s2"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and gpv1 :: "('c, 'out, 'in) gpv"   and gpv2 :: "('d, 'out, 'in) gpv" assumes "S s1 s2"   and "eq_\<I>_gpv A \<I> gpv1 gpv2" shows "rel_spmf (rel_sum (rel_prod A S) (\<lambda>(q, rpv1, rpv2) (q', rpv1', rpv2'). q = q' \<and> q' \<in> outs_\<I> \<I>' \<and> (\<exists>q''\<in>outs_\<I> \<I>. (\<forall>r\<in>responses_\<I> \<I>' q'. eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r'. r' \<in> responses_\<I> \<I> q'')) S) \<I>' (rpv1 r) (rpv1' r)) \<and> (\<forall>r'\<in>responses_\<I> \<I> q''. eq_\<I>_gpv A \<I> (rpv2 r') (rpv2' r'))))) (inline1 callee1 gpv1 s1) (inline1 callee2 gpv2 s2)"
similar_matrix_def: fixes A :: "(('d, 'e) vec, 'e) vec"   and B :: "(('d, 'f) vec, 'f) vec" shows "(A \<sim> B) = (\<exists>P. invertible P \<and> A = P\<^sup>-\<^sup>1 ** B ** P)"
assms_1: shows "x0 \<in> carrier (\<G> n)"
elem_limit_simp: fixes x :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set" shows "(x \<in> limit f X) = (\<exists>n. x \<in> funpower f n X)"
Enc_keys_clean_def: fixes S :: "msg set" shows "Enc_keys_clean S \<equiv> \<forall>X Y. Enc X Y \<in> parts S \<longrightarrow> Y \<in> range LtK \<union> payload"
prod_lens_comp_plus: fixes X\<^sub>2 :: "'d \<Longrightarrow> 'e"   and Y\<^sub>2 :: "'f \<Longrightarrow> 'e"   and X\<^sub>1 :: "'g \<Longrightarrow> 'd"   and Y\<^sub>1 :: "'h \<Longrightarrow> 'f" assumes "X\<^sub>2 \<bowtie> Y\<^sub>2" shows "X\<^sub>1 \<times>\<^sub>L Y\<^sub>1 ;\<^sub>L (X\<^sub>2 +\<^sub>L Y\<^sub>2) = X\<^sub>1 ;\<^sub>L X\<^sub>2 +\<^sub>L Y\<^sub>1 ;\<^sub>L Y\<^sub>2"
coarser_partitions_with_def: fixes new_el :: "'b"   and P :: "'b set set" shows "coarser_partitions_with new_el P = insert (insert {new_el} P) (insert_into_member new_el P ` P)"
atLeastAtMost_iff: fixes i :: "'a Word.word"   and l :: "'a Word.word"   and u :: "'a Word.word" shows "(i \<in> signed.atLeastAtMost l u) = (l \<le>s i \<and> i \<le>s u)"
at_pre_def: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>'\<alpha>\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<upsilon> x" shows "\<tau> \<Turnstile> \<delta> H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
M3_axioms_def: shows "M3_axioms \<equiv> (\<forall>x y. PP x y = (P x y \<and> x \<noteq> y)) \<and> (\<forall>x y. (x = y) = (\<forall>z. P z x = P z y))"
dim: fixes k :: "nat" assumes "k < length vs" shows "vs ! k \<in> carrier_vec n"
sup_compl_top_left2: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (- x \<squnion> y) = \<top>"
commut_normal: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group.group G"   and "subgroup H G"   and "N \<lhd> G" shows "H <#>\<^bsub>G\<^esub> N = N <#>\<^bsub>G\<^esub> H"
gamma3_as_euclid_space_fun: shows "gamma3 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R i + (0, g1 (a + (b - a) * x)))"
new_Addr_SomeD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat" assumes "new_Addr h = \<lfloor>a\<rfloor>" shows "h a = None"
assms: shows "prefix\<cdot>[:x:]\<cdot>vs = TT"
AAss1Red3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and a :: "'addr val"   and i :: "'addr val" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val a\<lfloor>Val i\<rceil> := e,s\<rangle> -ta\<rightarrow> \<langle>Val a\<lfloor>Val i\<rceil> := e',s'\<rangle>"
extf_ext_compat_list: fixes f :: "'s" shows "ext_compat_list (extf f)"
dom_sound: fixes i :: "nat"   and j :: "nat" assumes "local.dom i j" shows "dominate i j"
big_small_asymmetric: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g"   and "g \<in> l F f" shows "\<forall>\<^sub>F x in F. f x = (0::'b)"
f: fixes x :: "'a" assumes "x \<in> S" shows "f x \<in> T \<and> g (f x) = x"