hcju/isabelleps huggingface dataset:
intros: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'BExp" shows "\<Gamma> \<turnstile>\<^sub>b e \<in> \<Union> (to_total \<Gamma> ` bexp_vars e) "
ACI_nPlus_singleton_PLUS: fixes xs :: "'b rexp list"   and x :: "'b rexp" assumes "xs \<noteq> []"   and "sorted xs"   and "distinct xs"   and "\<forall>x\<in>{x} \<union> set xs. \<nexists>r s. x = Plus r s" shows "ACI_nPlus x (PLUS xs) = (if x \<in> set xs then PLUS xs else PLUS (insort x xs))"
psubset_imp_subset: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subset> B" shows "A \<subseteq> B"
index_uminus_vec_2: fixes v :: "'a vec" shows "dim_vec (- v) = dim_vec v"
is_poincare_line_x_axis: shows "is_poincare_line x_axis"
wf_REV: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp" assumes "alphabet.wf wf_atom n r" shows "alphabet.wf wf_atom n (REV r)"
set_plus_mono2: fixes C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and F :: "'b set" assumes "C \<subseteq> D"   and "E \<subseteq> F" shows "C + E \<subseteq> D + F"
resCap_cf_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and p :: "(nat \<times> nat) list" shows "resCap_cf_impl cf p \<equiv> case p of [] \<Rightarrow> return 0 | e # p \<Rightarrow> cf_get cf e \<bind> (\<lambda>cap. assert (distinct p) \<bind> (\<lambda>_. nfoldli p (\<lambda>_. True) (\<lambda>e cap. cf_get cf e \<bind> (\<lambda>cape. return (min cape cap))) cap))"
induct: fixes P1 :: "('a, 'b) form \<Rightarrow> bool"   and P3 :: "('a, 'b) form list \<Rightarrow> bool"   and P2 :: "('a, 'b) form list \<Rightarrow> bool"   and form :: "('a, 'b) form" assumes "\<And>x. P1 (At x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P1 (Compound x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Modal x1 x2)"   and "P1 ff"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 # x2)" shows "P1 form"
nested_prop_atoms\<^sub>\<nu>_def: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "nested_prop_atoms\<^sub>\<nu> \<phi> X = {\<psi>[X]\<^sub>\<nu> |\<psi>. \<psi> \<in> nested_prop_atoms \<phi>}"
VWO_antisym: fixes x :: "V"   and y :: "V" assumes "(x, y) \<in> VWO"   and "(y, x) \<in> VWO" shows "x = y"
finite_imp_foldSetD: fixes A :: "'c set"   and e :: "'d"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "finite A"   and "e \<in> D"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D" shows "\<exists>x. (A, x) \<in> foldSetD D f e"
Ioi_le_Ico: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.preorder less_eq less" shows "ord.greaterThan less a \<subseteq> ord.atLeast less_eq a"
run_definitions_3: fixes ls :: "'c \<Rightarrow> 'b"   and ss :: "'c \<Rightarrow> 'a" shows "is_lfair ls ss \<equiv> \<forall>l i. \<exists>j\<ge>i. \<not> can_step l (ss j) \<or> ls j = l"
component_of_map_component: fixes f :: "nat \<Rightarrow> nat"   and v :: "'t" shows "component_of_term (map_component f v) = f (component_of_term v)"
lm062: fixes Y :: "'d set" shows "totalRels {} Y = {{}}"
assms: shows "\<forall>F x s. eval\<Pi>\<^sub>1 F x s = eval\<Pi>\<^sub>1 F x dj"
is_iso_class_def: fixes F :: "'a set" shows "B.is_iso_class F \<equiv> \<exists>f. f \<in> F \<and> F = \<lbrakk>f\<rbrakk>\<^sub>B"
\<Gamma>_sound: fixes \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset multiset"   and D :: "'b literal multiset"   and E :: "'b literal multiset"   and I :: "'b set" assumes "sound_inference_system \<Gamma>"   and "Infer CC D E \<in> \<Gamma>"   and "I \<Turnstile>m CC"   and "I \<Turnstile> D" shows "I \<Turnstile> E"
simps_1: fixes \<gamma> :: "'b \<Rightarrow> 'a set" shows "\<gamma>_option \<gamma> None = {}"
prodrg_component: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme"   and i :: "'d" assumes "f \<in> carrier (r\<Pi>\<^bsub>I\<^esub> A)"   and "i \<in> I" shows "f i \<in> carrier (A i)"
coprime_dvd_aux: fixes m :: "nat"   and n :: "nat"   and na :: "nat"   and ma :: "nat"   and mb :: "nat"   and nb :: "nat" assumes "gcd m n = Suc 0"   and "na dvd n"   and "ma dvd m"   and "mb dvd m"   and "nb dvd n"   and "ma * na = mb * nb" shows "ma = mb"
mkarr_inj_on_Ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Ide C t"   and "monoidal_language.Ide C u"   and "free_monoidal_category.mkarr C t = free_monoidal_category.mkarr C u" shows "t = u"
restrict_to_singleton: fixes P :: "('a \<times> 'b) set"   and x :: "'a" shows "P || {x} = {x} \<times> P `` {x}"
FG_suffix_G: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<phi>)" shows "\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n G\<^sub>n \<phi>"
dickson_grading_dgrad_dummy: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.graded_dickson_powerprod plus minus zero lcs" shows "plus.dickson_grading plus (graded_dickson_powerprod.dgrad_dummy plus)"
matchC_LC_mono: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and theta' :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "matchC_LC theta c d"   and "theta \<subseteq> theta'" shows "matchC_LC theta' c d"
ShuffleLR_PlusR: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "ShuffleLR r (Plus s t) = Plus (ShuffleLR r s) (ShuffleLR r t)"
unit_factor_poly_def: fixes p :: "'a poly" shows "unit_factor p = [:unit_factor (Polynomial.lead_coeff p):]"
inv_unique: fixes y :: "cltn2"   and x :: "cltn2"   and y' :: "cltn2" assumes "y \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"   and "x \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> y' = \<one>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"   and "x \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "y \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "y' \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "y = y'"
DONE_def: shows "DONE = Resumption.resumption.Done \<circ> Some"
Riemann_localization_integral: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and d :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "g absolutely_integrable_on {- pi..pi}"   and "0 < d"   and "\<And>x. \<bar>x\<bar> < d \<Longrightarrow> f x = g x" shows "(\<lambda>n. (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f x) - (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * g x)) \<longlonglongrightarrow> 0"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
one_add_one: shows "(1::'a) + (1::'a) = (2::'a)"
cycle_def: fixes xs :: "Vertices list" shows "cycle xs \<equiv> path xs \<and> 2 < length xs \<and> last xs \<rightarrow>\<^bsub>T2\<^esub> hd xs"
GE_resOf_def: shows "GE_resOf \<equiv> ProblemIkTpartG.GE_resOf resOf"
size_filter_mset_lesseq: fixes f :: "'b \<Rightarrow> bool"   and M :: "'b multiset" shows "size (filter_mset f M) \<le> size M"
WHILEIT_weaken: fixes I :: "'a \<Rightarrow> bool"   and I' :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm"   and x :: "'a" assumes "\<And>x. I x \<Longrightarrow> I' x" shows "WHILEIT I' b f x \<le> WHILEIT I b f x"
inverse_inverse_eq: fixes a :: "'a" shows "inverse (inverse a) = a"
refl_onP_DomainPD: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "refl_onP A r" shows "A = {a. Domainp r a}"
filterlim_divide_at_bot_at_top_iff_1: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> 0" shows "(LIM x F. f x / g x :> at_bot) = ((g \<longlongrightarrow> 0) F \<and> (g has_sgnx - sgn c) F)"
comp_reduce: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and k :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "CC.comp f g = k"   and "CC.seq f g"   and "CC.seq g h" shows "CC.comp f (CC.comp g h) = CC.comp k h"
simps_1: fixes literal :: "Literal"   and newWl :: "nat list"   and state :: "State" shows "notifyWatches_loop literal [] newWl state = state\<lparr>getWatchList := (getWatchList state)(literal := newWl)\<rparr>"
abs_eq: fixes x :: "('b \<times> real) nonzero" shows "chart_last_domainP (Proj x) = chart_last_domain_nonzeroP x"
res': shows "bezout_coefficients_i ff_ops (of_int_poly_i ff_ops f) (of_int_poly_i ff_ops g) = (a', b')"
termination_0_1_immediate: fixes p :: "real"   and s :: "bool \<times> nat" assumes "\<And>s. fst s \<Longrightarrow> p \<le> spmf (map_spmf fst (case s of (b, x) \<Rightarrow> map_spmf (\<lambda>b'. (\<not> b', x + (if b' then 0 else 1))) (bernoulli p))) False"   and "0 < p"   and "\<And>s. fst s \<Longrightarrow> lossless_spmf (case s of (b, x) \<Rightarrow> map_spmf (\<lambda>b'. (\<not> b', x + (if b' then 0 else 1))) (bernoulli p))" shows "lossless_spmf (local.while s)"
vec_component: fixes x :: "'d"   and i :: "'e" shows "vec x $ i = x"
full_deriv_wSts_trancl_weighted_RP: shows "full_chain (\<leadsto>\<^sub>w\<^sup>+) wSts"
reads_def: fixes S :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and getter :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" shows "reads S getter h h' = ((\<forall>P\<in>S. reflp P \<and> transp P) \<and> ((\<forall>P\<in>S. P h h') \<longrightarrow> preserved getter h h'))"
that: fixes a :: "'a"   and b :: "'a" assumes "x = a + b"   and "a \<in> A"   and "b \<in> B" shows "thesis"
bot_ideal: fixes X :: "'a set" assumes "X \<in> ord.ideals (\<lambda>x y. y \<le> x)" shows "\<top> \<in> X"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'g list"   and x :: "'g" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
all_conj_distrib: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
basic_cqe_conv_2b: fixes p :: "complex poly" shows "(\<exists>x. poly p x \<noteq> 0) = (p \<noteq> 0)"
f1: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < b"   and "b < c"   and "c < d" shows "f1 {a, b, c, d} = f {(a, c), (b, d)}"
greedy_path_conforms: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V"   and "strategy p \<sigma>"   and "strategy p** \<sigma>'" shows "path_conforms_with_strategy p (greedy_conforming_path p \<sigma> \<sigma>' v0) \<sigma>"
bigI: fixes c :: "real"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "0 < c"   and "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)" shows "f \<in> \<Omega>[F](g)"
2: shows "deg (R\<lparr>carrier := S\<rparr>) g \<le> deg R g"
prv_imp_monoL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<chi> \<psi>)" shows "bprv (imp (imp \<phi> \<chi>) (imp \<phi> \<psi>))"
component_of_term_of_pair: fixes t :: "'a"   and k :: "nat" shows "component_of_term (term_of_pair (t, k)) = k"
genby_genset_closed: fixes s :: "'a"   and S :: "'a set" assumes "s \<in> S" shows "s \<in> \<langle>S\<rangle>"
lm060: fixes P :: "('c \<times> 'd) set" shows "inj_on fst P = inj_on (snd \<circ> flip) P"
induct: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) set \<times> bool"   and x1 :: "'b"   and x2 :: "'c"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "whileLoop_terminates C B x1 x2"   and "\<And>r s. \<not> C r s \<Longrightarrow> P r s"   and "\<And>r s. \<lbrakk>C r s; \<forall>(r', s')\<in>fst (B r s). whileLoop_terminates C B r' s' \<and> P r' s'\<rbrakk> \<Longrightarrow> P r s" shows "P x1 x2"
nn_transfer_operator_cong: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "AE x in M. f x = g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_transfer_operator f x = nn_transfer_operator g x"
ZFfun: fixes f :: "ZF" assumes "isZFfun f" shows "f = ZFfun (|dom|f) (|cod|f) ((|@|) f)"
greatest_def: fixes P :: "'b \<Rightarrow> bool"   and po :: "'b potype" shows "greatest P po = (SOME x. x \<in> pset po \<and> P x \<and> (\<forall>y\<in>pset po. P y \<longrightarrow> (y, x) \<in> order po))"
induct: fixes x :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and P :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> \<Rightarrow> bool" assumes "hereditarily_fs x"   and "\<And>tset\<^sub>\<alpha>. \<lbrakk>finite (supp tset\<^sub>\<alpha>); \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>; \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"   and "\<And>f \<phi>. P (Pred\<^sub>\<alpha> f \<phi>)"   and "\<And>t\<^sub>\<alpha> f \<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>)" shows "P x"
cv_hom_par: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<parallel> Multirelations.U \<parallel> y \<parallel> Multirelations.U = x \<parallel> y \<parallel> Multirelations.U"
S3RPC_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> ($S3 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> <RPCNext crCh rmCh rst p>_(r p) \<longrightarrow> (S4 rmhist p \<or> S6 rmhist p)$"
keys_higherPoly\<^sub>0: fixes b :: "nat"   and i :: "nat"   and mp :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "keys (higherPoly\<^sub>0 b i mp) = lowerPowers b i ` keys mp"
simps_2: shows "hmset_of_enat \<infinity> = \<omega>"
snd_scaleR: fixes r :: "real"   and A :: "'d \<times> 'c" shows "snd (r *\<^sub>R A) = r *\<^sub>R snd A"
hyps_3: assumes "\<not> lnull (ltl P)"   and "path_conforms_with_strategy p (ltl P) \<sigma>"   and "\<not> lnull P'"   and "path_conforms_with_strategy p P' \<sigma>"   and "llast (ltl P) \<in> VV p \<Longrightarrow> \<sigma> (llast (ltl P)) = lhd P'" shows "path_conforms_with_strategy p (lappend (ltl P) P') \<sigma>"
list_all2_rev1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
linepath_image_01: fixes a :: "'b"   and b :: "'b" shows "linepath a b ` {0..1} = {a--b}"
x: shows "O x (\<sigma>v. \<not> O v x) = (\<exists>v. \<not> O v x \<and> O x v)"
is_filter_Rep_filter: fixes F :: "'b filter" shows "is_filter (Rep_filter F)"
1: shows "coreflexive ((w \<sqinter> - d \<sqinter> - d\<^sup>T) * (w\<^sup>T \<sqinter> - d \<sqinter> - d\<^sup>T))"
assms_2: shows "g \<in> F n \<rightarrow>\<^sub>M N"
ll: shows "Partial_Cost_Model.steps R (xs @ [x, x]) (OPT2 (xs @ [x, x]) R) = [x, y]"
mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> L F g1"   and "f2 \<in> L F g2" shows "(\<lambda>x. f1 x * f2 x) \<in> L F (\<lambda>x. g1 x * g2 x)"
sender_ip_valid: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), a, uu_). anycast (\<lambda>m. not_Pkt m \<longrightarrow> msg_sender m = i) a)"
rbt_inter_with_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "rbt_inter_with f = rbt_inter_with_key (\<lambda>_. f)"
AgentY_X_PPA: shows "\<lfloor>PPA (Agent X)\<rfloor>\<^sub>Y"
assms_1: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
focus_monomial: fixes X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c" shows "focus X (monomial c t) = monomial (monomial c (except t X)) (except t (- X))"
equalityD1: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" shows "A \<subseteq> B"
T_ins_tree_length: fixes t :: "'b tree"   and ts :: "'b tree list" shows "T_ins_tree t ts + length (ins_tree t ts) = 2 + length ts"
image_mset_cong: fixes M :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "image_mset f M = image_mset g M"
Pruned_Port_Graph_axioms: shows "Pruned_Port_Graph task.nodes task.inPorts task.outPorts vertices nodeOf task_edges"
is_generator_set: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and canonical_basis :: "'a list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "cspan (set canonical_basis) = UNIV"
pcr_cr_eq: shows "pcr_fin_string = cr_fin_string"
intro: fixes f :: "'q \<Rightarrow> 'r" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>x. f (- x) = - f x"   and "\<And>x y. f (x * y) = f x * f y"   and "f (0::'q) = (0::'r)"   and "f (1::'q) = (1::'r)" shows "ring_homomorphism f"
completeI: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x" shows "complete A r"
image_add_atLeastLessThan: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "'a"   and i :: "'a"   and j :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "plus k ` ord.atLeastLessThan less_eq less i j = ord.atLeastLessThan less_eq less (plus i k) (plus j k)"
that: fixes B :: "real set" assumes "0 \<in> B"   and "open B"   and "B \<subseteq> T"   and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> 0 s)) at 0 within B" shows "thesis"
itop_carrier: fixes A :: "'b set" shows "carrier\<^bsub>topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s})\<^esub> = A\<^sup>\<omega>"
simps_1: fixes ys :: "'b llist" shows "prepend [] ys = ys"
assms_2: shows "[p \<^bold>\<rightarrow> \<lparr>R,x,y\<rparr> in v]"
remdups_sort_def: fixes xs :: "'c list" shows "remdups_sort xs = remdups_adj (sort xs)"
dist_on_eq: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list"   and V :: "nat set" assumes "is_shortest_path_on G m n p V" shows "dist_on G m n V = path_weight G p"
simps_2: fixes n :: "nat" shows "enumerate_aexp_inputs (V (I n)) = {n}"
p'_def: fixes t :: "real" shows "ego2.p' t = v\<^sub>e + a\<^sub>e * t"
pref_eq_true: fixes P :: "'b \<Rightarrow> bool" assumes "|~ P" shows "|~ P = #True"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<eta>' \<epsilon>'"
targetsD_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "b \<in> weak_composition.targets V H \<mu>" shows "H b \<mu> \<noteq> partial_magma.null V"
freeword_funlift_Abs_freeword_Cons: fixes x :: "'a \<times> bool"   and xs :: "('a \<times> bool) list"   and f :: "'a \<Rightarrow> 'b" assumes "proper_signed_list (x # xs)" shows "freeword_funlift f (Abs_freeword (x # xs)) = apply_sign f x + freeword_funlift f (Abs_freeword xs)"
FAss_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<bullet>F{D} := e') h s s'"
mult_le_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c * a \<le> c * b) = (((0::'a) < c \<longrightarrow> a \<le> b) \<and> (c < (0::'a) \<longrightarrow> b \<le> a))"
first_bothNetsd: fixes x :: "('b, 'c) Combinators" assumes "\<not> member DenyAll x" shows "first_bothNet x = {first_srcNet x, first_destNet x}"
simps_19: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "char list exp" shows "fv (INIT C (Cs,b) \<leftarrow> e) = fv e"
mod_mult_right_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a (modulo b c)) c = modulo (times a b) c"
code: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "siterate f x = x ## siterate f (f x)"
implSecure_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> valid \<union> range LtK \<union> Tags"   and "implSecure A B M \<in> synth (analz H)" shows "implSecure A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
intros_32: fixes n :: "nat"   and es :: "e list" shows "\<lparr>[Label n es [Trap]]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
rhs_coeffs: shows "((1 + fps_X ^ p) ^ (n div p) * (1 + fps_X) ^ (n mod p)) $ k = int ((n div p choose k div p) * (n mod p choose k mod p))"
prefix_match_semantics_simple_match: fixes m :: "'b word \<times> nat"   and pm :: "'b prefix_match" assumes "toprefixmatch m = Some pm"   and "valid_prefix pm" shows "prefix_match_semantics pm = simple_match_ip m"
Cons_3: shows "length (y # ys) = length (x # xs)"
inj: fixes j :: "nat" assumes "j < n \<or> j < i"   and "cis (real i * 2 * pi / real n) = cis (real j * 2 * pi / real n)" shows "i = j"
igSwapIGAbsSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSwapIGAbsSTR MOD \<equiv> \<forall>zs z1 z2 xs x X. igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
self_append_conv2: fixes ys :: "'e list"   and xs :: "'e list" shows "(ys = xs @ ys) = (xs = [])"
ceiling_less_neg_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lceil>x\<rceil> < - numeral v) = (x \<le> - numeral v - (1::'b))"
Inf_set_def: fixes A :: "'c set set" shows "\<Inter> A = {x. Inf ((\<in>) x ` A)}"