hcju/isabelleps huggingface dataset:
LT23: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ (P \<longrightarrow> \<circle>Q) \<longrightarrow> P \<longrightarrow> \<diamond>Q"
asymp_equiv_transfer: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<sim>[F] g1"   and "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "f2 \<sim>[F] g2"
pred_alt_def_1: fixes name :: "name"   and vs :: "value list" shows "vwellformed (Vconstr name vs) = (True \<and> list_all vwellformed vs)"
convex_alt: fixes s :: "'c set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
path_approx_polynomial_function: fixes g :: "real \<Rightarrow> 'c"   and e :: "real" assumes "path g"   and "0 < e" obtains p :: "real \<Rightarrow> 'c" where "polynomial_function p" and "pathstart p = pathstart g" and "pathfinish p = pathfinish g" and "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (p t - g t) < e"
cauchy_imp_bounded: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
part: shows "partition ((=) (length bef)) (\<sigma> (f, Suc (length bef + length aft))) = (lb, nlb)"
sel_specI: fixes sel :: "(('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list \<Rightarrow> (('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list \<Rightarrow> ((('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list \<Rightarrow> nat \<times> 'e \<Rightarrow> ((('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list" assumes "\<And>gs bs ps data. ps \<noteq> [] \<Longrightarrow> sel gs bs ps data \<noteq> [] \<and> set (sel gs bs ps data) \<subseteq> set ps" shows "sel_spec sel"
CauchysMeanTheorem: fixes z :: "real list" assumes "pos z" shows "gmean z \<le> mean z"
span_explicit: fixes b :: "'c set" shows "vs2.span b = {\<Sum>a\<in>t. r a *b a |t r. finite t \<and> t \<subseteq> b}"
ivl_rel_def: fixes S :: "('a \<times> 'b) set" shows "\<langle>S\<rangle>ivl_rel = (S \<times>\<^sub>r S) O br set_of_ivl top"
process_prop_6_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_6 P \<equiv> \<forall>xs X. xs \<in> snd P \<longrightarrow> (xs, X) \<in> fst P"
valid_cfgI: fixes s :: "'s \<times> ('c \<Rightarrow> real) set"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "s \<in> \<S>"   and "cfg \<in> R_G.cfg_on s" shows "cfg \<in> R_G.valid_cfg"
simps: fixes as :: "nat list"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "substW as \<sigma> = subst (w2rexp as) \<sigma>"
distinct_2: fixes x21 :: "Unary_PCF.type"   and x22 :: "Unary_PCF.type" shows "x21 \<rightarrow> x22 \<noteq> \<B>"
suminf_SUP_eq_directed: fixes I :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "\<And>N i j. \<lbrakk>i \<in> I; j \<in> I; finite N\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f i n \<le> f k n \<and> f j n \<le> f k n"   and "\<And>n i. 0 \<le> f n i" shows "(\<Sum>i. SUP n\<in>I. f n i) = (SUP n\<in>I. \<Sum>i. f n i)"
simps_1: fixes v\<^sub>1 :: "val"   and v\<^sub>2 :: "val" shows "binop (Eq, v\<^sub>1, v\<^sub>2) = \<lfloor>Bool (v\<^sub>1 = v\<^sub>2)\<rfloor>"
HPT_SAStates_Target_Source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and A :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and T :: "'d" assumes "TS \<in> HPT ST"   and "A \<in> SAs (HA ST)"   and "S \<in> States A"   and "T \<in> States A"   and "S \<in> Conf ST"   and "T \<in> Target TS" shows "S \<in> Source TS"
arev_eq_iff: fixes G :: "('a, 'b) pre_digraph"   and arev :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "bidirected_digraph G arev" shows "(arev x = arev y) = (x = y)"
comp_pres_disj: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> disjunctive.disjunctive sup sup times"   and "y \<in> disjunctive.disjunctive sup sup times" shows "times x y \<in> disjunctive.disjunctive sup sup times"
silent_moves_no_slice_edges: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')"   and "tl ms = targetnodes rs"   and "length rs = length cs"   and "\<forall>i<length cs. call_of_return_node (tl ms ! i) (sourcenode (cs ! i))" shows "slice_edges S cs as = [] \<and> (\<exists>rs'. tl ms' = targetnodes rs' \<and> length rs' = length (upd_cs cs as) \<and> (\<forall>i<length (upd_cs cs as). call_of_return_node (tl ms' ! i) (sourcenode (upd_cs cs as ! i))))"
pheap_insert: fixes h :: "'b heap"   and x :: "'b" assumes "pheap h" shows "pheap (insert x h)"
fiter3: fixes c :: "'a::refinement_lattice" shows "((c::'a::refinement_lattice)\<^sup>\<star>)\<^sup>\<star> = c\<^sup>\<star>"
that: fixes u :: "'a list"   and a :: "'a"   and v :: "'a stream" assumes "w = u @- a ## v"   and "degen cs (a, fold (count cs) u k)" shows "thesis"
c_pair_def: fixes x :: "nat"   and y :: "nat" shows "c_pair x y = sf (x + y) + x"
converse_pseudonatural_equivalence_axioms: shows "converse_pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
typing_Ax_elim2: fixes x :: "name"   and B :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and a :: "coname"   and \<Delta> :: "(coname \<times> ty) list" assumes "((x, B) # \<Gamma>) \<turnstile> Ax x a \<turnstile> \<Delta>" shows "(a, B) \<in> set \<Delta>"
index_last_take: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs ! i = last (take (Suc i) xs)"
smult_1_left: fixes p :: "'c poly" shows "Polynomial.smult (1::'c) p = p"
parent_sum: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat"   and F :: "(nat \<times> int) list \<Rightarrow> 'a" assumes "p \<in> grid (child b dir d) {d}"   and "d < length b" shows "sum F (parents d b p) = F b + sum F (parents d (child b dir d) p)"
succ_upRule: fixes Ps :: "'a list"   and \<Phi> :: "'b form multiset"   and \<Psi> :: "'b form multiset" assumes "(Ps,  \<Phi> \<Rightarrow>* \<Psi>) \<in> upRules" shows "\<Psi> = \<Empt> \<or> (\<exists>A. \<Psi> = \<LM> A  \<RM>)"
LIMSEQ_cong: fixes f :: "nat \<Rightarrow> real"   and x :: "real"   and g :: "nat \<Rightarrow> real" assumes "f \<longlonglongrightarrow> x"   and "\<forall>\<^sup>\<infinity>n. f n = g n" shows "g \<longlonglongrightarrow> x"
igWlsAllDisj_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igWlsAllDisj MOD \<equiv> igWlsDisj MOD \<and> igWlsAbsDisj MOD"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "('a, 'b) ffun" shows "pfun_of (x \<inter>\<^sub>f xa) = pfun_of x \<inter>\<^sub>p pfun_of xa"
assms_4: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1" shows "denote x2 (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote x2 \<rho>"
m_lcomm: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<in> carrier N"   and "y \<in> carrier N"   and "z \<in> carrier N" shows "x \<oplus>\<^bsub>N\<^esub> (y \<oplus>\<^bsub>N\<^esub> z) = y \<oplus>\<^bsub>N\<^esub> (x \<oplus>\<^bsub>N\<^esub> z)"
cong_add_rcancel: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b"   and y :: "'b"   and n :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (plus x a) (plus y a) n = unique_euclidean_semiring.cong modulo x y n"
nxt_stable: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and b' :: "nat" assumes "total_cons s"   and "eval (total_cons.r_next s) [b] \<down>= b'" shows "\<forall>x<e_length b. e_nth b x = e_nth b' x"
rangeC_eq_empty: fixes A :: "('b \<Rightarrow> 'a) set" shows "(rangeC A = {}) = (A = {})"
iwlsFSbSwTR_alphaAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSbSwTR MOD" shows "(qGood qX \<longrightarrow> (\<forall>qX'. qX #= qX' \<longrightarrow> qInit MOD qX = qInit MOD qX')) \<and> (qGoodAbs qA \<longrightarrow> (\<forall>qA'. qA $= qA' \<longrightarrow> qInitAbs MOD qA = qInitAbs MOD qA'))"
empty: fixes ns :: "('a \<times> 'a) set" shows "({#}, {#}) \<in> multpw ns"
Sup_distr: fixes X :: "('b \<times> 'b) set set"   and y :: "('b \<times> 'b) set" shows "\<Union> X O y = \<Union> {z. \<exists>x\<in>X. z = x O y}"
RSD_pareto_eqclass_def: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "RSD_pareto_eqclass agents alts R A = (A \<noteq> {} \<and> A \<subseteq> alts \<and> (\<forall>x\<in>A. \<forall>y\<in>alts. (y \<in> A) = (\<forall>i\<in>agents. R i x y)))"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_bef_erase_b (s, l, r) lm n ires = (\<exists>ln m rn. l = Bk \<up> ln @ Bk # Bk # ires \<and> r = Bk # Oc \<up> m @ Bk # <drop (s div 2) lm> @ Bk \<up> rn)"
lrev_work3_lrev_work2_eq': shows "lrev = wrapH\<cdot>lrev_work3"
assms_4: shows "last (msgs (S t i) cid) = Marker"
zero_le: fixes x :: "'i multiset" shows "{#} \<subseteq># x"
post_meas0_bell11_snd: shows "post_meas0 2 |\<beta>\<^sub>1\<^sub>1\<rangle> 1 = - |unit_vec 4 2\<rangle>"
IH: shows "hereditarily_fs (p \<bullet> t\<^sub>\<alpha>_)"
rep_eq: shows "\<lbrakk>\<top>\<rbrakk>\<^sub>e = (\<lambda>A. \<top>)"
bhv_hn_context: fixes tid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "honest tid"   and "\<parallel>tid\<parallel>\<^bsub>t n\<^esub>"   and "\<exists>n'. latestAct_cond tid t n n'" shows "\<exists>nid'. \<parallel>nid'\<parallel>\<^bsub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> (mining (\<sigma>\<^bsub>tid\<^esub>t n) \<and> (\<exists>b. bc (\<sigma>\<^bsub>tid\<^esub>t n) = bc (\<sigma>\<^bsub>nid'\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>) @ [b]) \<or> \<not> mining (\<sigma>\<^bsub>tid\<^esub>t n) \<and> bc (\<sigma>\<^bsub>tid\<^esub>t n) = bc (\<sigma>\<^bsub>nid'\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>))"
psums_decompose: shows "psums xs = replicate p (0::'a) @ psums (xs ! p # xs2)"
scale_right_imp_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'a"   and b :: "'a" assumes "vector_space scale"   and "x \<noteq> (0::'b)"   and "scale a x = scale b x" shows "a = b"
rec_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" shows "rec MOD \<equiv> fromIMor (iter (fromMOD MOD))"
HPair_def: fixes a :: "tm"   and b :: "tm" shows "HPair a b = Eats (Eats Zero (Eats (Eats Zero b) a)) (Eats (Eats Zero a) a)"
repc'_def: shows "repc' \<equiv> \<lambda>cfg'. repcs (THE s. s \<in> set_pmf (rept s (action cfg)) \<and> abss s = state cfg') cfg'"
section_is_mono: fixes g :: "'b" assumes "D.section g" shows "D.mono g"
distinct_11: fixes x2 :: "'c"   and x3 :: "('b, 'c) fm" shows "Nom x2 \<noteq> (\<^bold>\<not> x3)"
load_tmps_distinct_nth_update: fixes i :: "nat"   and "is" :: "instr list"   and p :: "'a"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "distinct_load_tmps is" shows "load_tmps_distinct (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
pw_ords_iff_1: fixes S :: "'a nres"   and S' :: "'a nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
eval_peval: fixes h :: "char list \<Rightarrow> bool"   and g :: "'b pfm \<Rightarrow> bool"   and p :: "'b fm" shows "eval h (g \<circ> lift) p = peval h g (lift p)"
ballE: fixes A :: "'k set"   and P :: "'k \<Rightarrow> bool"   and x :: "'k" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
simps: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" shows "ord.lexordp_eq less a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp_eq less xs ys))"
disc_4: fixes x11 :: "name"   and x12 :: "value list" shows "\<not> is_Vabs (Vconstr x11 x12)"
hcomp_in_vhom: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a"   and \<nu> :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow> k\<guillemotright>"   and "src h = trg f" shows "\<guillemotleft>\<nu> \<star> \<mu> : h \<star> f \<Rightarrow> k \<star> g\<guillemotright>"
nonempty_\<Sigma>: shows "\<Sigma> \<noteq> {}"
omega_absorb1: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y \<cdot> ad ( |x\<rangle> y) = \<Omega> x y"
idiff_0_right: fixes n :: "enat" shows "n - 0 = n"
exhaust: fixes y :: "graph" obtains x1 :: "face list" and x2 :: "nat" and x3 :: "face list list" and x4 :: "nat list" where "y = Graph x1 x2 x3 x4"
integrable_neg: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f integrable_on S" shows "(\<lambda>x. - f x) integrable_on S"
extended_Gromov_product_at_diff1: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and f :: "'a" shows "extended_Gromov_product_at e x y \<le> extended_Gromov_product_at f x y + ereal (dist e f)"
Suc: shows "four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) B ^\<^sub>m k = four_block_mat (A ^\<^sub>m k) (0\<^sub>m n m) (0\<^sub>m m n) (B ^\<^sub>m k)"
closeSubst_def: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" shows "closeSubst Rel \<equiv> {(\<Psi>, P, Q) |\<Psi> P Q. \<forall>\<sigma>. wellFormedSubst \<sigma> \<longrightarrow> (\<Psi>, P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel}"
integrable_mult_left: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x * c)"
ltake_LNil: fixes i :: "nat" shows "Co_Snapshot.ltake i LNil = []"
field_differentiable_transform_within: fixes d :: "real"   and x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "0 < d"   and "x \<in> S"   and "\<And>x'. \<lbrakk>x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"   and "f field_differentiable at x within S" shows "g field_differentiable at x within S"
cPar2B: fixes a :: "subject"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "!P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a\<guillemotleft>x\<guillemotright> \<prec> P')" shows "F C (P \<parallel> !P) (a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> P')"
subgraph_refl: fixes G :: "('d, 'e) labeled_graph" shows "subgraph G G = graph G"
get_elements_by_class_name_def: shows "get_elements_by_class_name \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_class_name to_tree_order get_attribute"
gen_model_semantic_equivalence: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w' :: "'g"   and w :: "'g"   and \<phi> :: "('e, 'f) Kform" assumes "kripke M"   and "w' \<in> worlds (gen_model M w)" shows "M, w' \<Turnstile> \<phi> = gen_model M w, w' \<Turnstile> \<phi>"
prime_divisor_exists: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a one" shows "\<exists>b. dvd.dvd times b a \<and> normalization_semidom.prime zero times one normalize b"
injectivity: fixes x :: "'b"   and y :: "'b" assumes "inv hom x = inv hom y" shows "x = y"
split_indicator: fixes P :: "'c \<Rightarrow> bool"   and S :: "'d set"   and x :: "'d" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'c)) \<and> (x \<notin> S \<longrightarrow> P (0::'c)))"
index_lt_replication: shows "\<Lambda> < \<Lambda> * (\<v> - 1) div (\<k> - 1)"
dsem_to_ssem: fixes \<theta> :: "('a, 'b) trm"   and I :: "('a, 'c, 'b) interp"   and \<nu> :: "(real, 'b) vec \<times> (real, 'b) vec" assumes "dfree \<theta>" shows "dterm_sem I \<theta> \<nu> = sterm_sem I \<theta> (fst \<nu>)"
dyn_ambiguous: 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 C :: "char list"   and M :: "char list"   and Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list" assumes "\<forall>mthd Cs'. \<not> P \<turnstile> C has least M = mthd via Cs'"   and "P \<turnstile> (C, Cs) has overrider M = mthd via Cs'" shows "P \<turnstile> (C,Cs) selects M = mthd via Cs'"
select_convs_1: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "iinput \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = iinput"
assms: shows "pdevs_apply X i \<notin> set (dense_list_of_pdevs X)"
lunit_char_1: fixes f :: "'a" assumes "Left.ide f" shows "Left.in_hom (Left.lunit f) (Left.L f) f"
star_unfoldl: fixes x :: "'a" shows "1' + x\<^sup>+ \<le> x\<^sup>\<star>"
assms: shows "C \<in> Sup_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)"
gt_sym_trans: fixes h :: "'s"   and g :: "'s"   and f :: "'s" assumes "h >\<^sub>s g"   and "g >\<^sub>s f" shows "h >\<^sub>s f"
not_choice_simps_5: fixes l :: "'d"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}broadcast(fmsg) . p)"
zero_less_norm_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(0 < norm x) = (x \<noteq> zero)"
C_Let: fixes E :: "trm \<Rightarrow> trm"   and p :: "pat"   and u :: "trm" assumes "E \<in> ctxt" shows "(\<lambda>t. LET p = E t IN u) \<in> ctxt"
singular_chain_of: fixes p :: "nat"   and X :: "'b topology"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "singular_chain p X (frag_of c) = singular_simplex p X c"
valid_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes P :: "'state"   and \<alpha> :: "'act"   and t\<^sub>\<alpha> :: "('a, 'pred, 'act) Tree\<^sub>\<alpha>" shows "valid_Tree\<^sub>\<alpha> P (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>) = (\<exists>\<alpha>' t\<^sub>\<alpha>' P'. Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> \<alpha>' t\<^sub>\<alpha>' \<and> P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> valid_Tree\<^sub>\<alpha> P' t\<^sub>\<alpha>')"
bDkA: shows "(@\<^sub>v) (0\<^sub>v n) ` baseD \<subseteq> mat_kernel A"
Noto_undef: fixes \<phi> :: "fml option" shows "(Noto \<phi> = undeff) = (\<phi> = undeff)"
zcf_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "zcf (p \<otimes>\<^bsub>P\<^esub> q) = zcf p \<otimes> zcf q"
assms_5: shows "prv (imp \<phi>1 \<chi>)"
ereal_uminus_divide: fixes x :: "ereal"   and y :: "ereal" shows "- x / y = - (x / y)"
freeword_funlift_im': fixes xs :: "('a \<times> bool) list"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "freeword_funlift f (Abs_freeword xs) \<in> \<langle>f ` S\<rangle>"
ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "j < length ts\<^sub>s\<^sub>b"   and "i \<noteq> j"   and "ts\<^sub>s\<^sub>b ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts\<^sub>s\<^sub>b ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
m_1: shows "(p ^^ m) x = (p ^^ n) x"
carrier_is_subring: shows "subring (carrier (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)) (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)"
surj_pair: fixes p :: "'g \<times> 'h" shows "\<exists>x y. p = (x, y)"
clw_rel_def: fixes A :: "('e \<times> 'f set) set" shows "clw_rel A = lw_rel O \<langle>A\<rangle>set_rel O br \<Union> top"
psimps_4: fixes M :: "'b trm"   and N :: "'b trm"   and v :: "'b" assumes "unify_dom (M \<cdot> N, Var v)" shows "unify (M \<cdot> N) (Var v) = (if Var v \<prec> M \<cdot> N then None else Some [(v, M \<cdot> N)])"