hcju/isabelleps huggingface dataset:
Conseq: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set"   and P :: "('a, 'b, 'c) ann"   and c :: "('a, 'b, 'c) com"   and Q :: "'a set"   and A :: "'a set" assumes "\<exists>P' Q' A'. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (weaken_pre P P') c Q', A' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A"
comp_apply: fixes f :: "'c::type \<Rightarrow> 'b::type"   and g :: "'d::type \<Rightarrow> 'c::type"   and x :: "'d::type" shows "((f::'c::type \<Rightarrow> 'b::type) \<circ> (g::'d::type \<Rightarrow> 'c::type)) (x::'d::type) = f (g x)"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'b"   and as :: "'b list" shows "bisum f (a # as) = f (0::'b) a + bisumTail f (a # as)"
alt_nondetT_def: shows "alt = nondetM_base.alt_nondet Monad_Overloading.return Monad_Overloading.bind (+)"
simps_1: fixes ao :: "'a option" shows "Compl_set_less_eq_aux ao [] [] = (\<not> proper_interval ao None)"
lindep_span: fixes U :: "'c set" assumes "U \<subseteq> carrier V"   and "finite U" shows "lin_dep U = (\<exists>u\<in>U. u \<in> span (U - {u}))"
elims_2: fixes x :: "'b list \<times> 'b list" assumes "invar x" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "fs = [] \<longrightarrow> rs = []"
l_comb_transpos1: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {l. l \<le> Suc n} \<rightarrow> A"   and "f \<in> {l. l \<le> Suc n} \<rightarrow> H"   and "j < Suc n" shows "l_comb R M (Suc n) s f = l_comb R M (Suc n) (cmp s (transpos j (Suc n))) (cmp f (transpos j (Suc n)))"
not_le_imp_less: fixes y :: "complex"   and x :: "complex" assumes "\<not> y \<le>\<^sub>\<complex> x" shows "x <\<^sub>\<complex> y"
finiteListDecomposeSet: fixes L :: "'b list set" assumes "finite L" shows "finite {(a, b). \<exists>l. l \<in> L \<and> l = a @ b}"
cnj_E2: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A AND B" shows "H \<turnstile> B"
hyps_1: assumes "{$$} \<turnstile>\<^sub>W e : \<tau>" shows "\<Gamma> \<turnstile>\<^sub>W e : \<tau>"
homotopic_paths_imp_path: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q" shows "path p \<and> path q"
\<beta>\<gamma>_1: shows "elts \<beta> \<subseteq> elts \<alpha>"
simps_17: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set" shows "(case Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
ldistinct_coinduct: fixes X :: "'c llist \<Rightarrow> bool"   and xs :: "'c llist" assumes "X xs"   and "\<And>xs. \<lbrakk>X xs; \<not> lnull xs\<rbrakk> \<Longrightarrow> lhd xs \<notin> lset (ltl xs) \<and> (X (ltl xs) \<or> ldistinct (ltl xs))" shows "ldistinct xs"
prems_4: shows "\<not> ListMem h (a # l2a)"
pdec2_encode_Pr: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "pdec2 (encode (Pr n f g)) = triple_encode n (encode f) (encode g)"
inline1_fixp_induct: fixes P :: "(('c, 'call, 'ret) gpv \<Rightarrow> 's \<Rightarrow> ('c \<times> 's + 'call' \<times> ('ret \<times> 's, 'call', 'ret') rpv \<times> ('c, 'call, 'ret) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. P inline1' \<Longrightarrow> P (\<lambda>gpv s. the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))" shows "P local.inline1"
sameness: fixes i :: "nat"   and j :: "nat" shows "(CHR ''a'' ^^^ i @ CHR ''b'' ^^^ j \<in> A) = (i = j)"
dagger_trans_eq: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod (t_star x) (t_star x) = t_star x"
set_take_two: fixes A :: "'a set" assumes "2 \<le> card A" obtains x :: "'a" and y :: "'a" where "x \<in> A" and "y \<in> A" and "x \<noteq> y"
TransApp_subst: fixes a :: "'a"   and e :: "exp"   and v :: "Vars.var"   and x :: "Vars.var"   and y :: "Vars.var" shows "(TransApp a e v)[x::=y] = TransApp a e[x::=y] v[x::v=y]"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs G) = (edge_rev M a \<noteq> a)"
iRelease_False_left: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool" shows "(False. t' \<R> t I. Q t) = (\<box> t I. Q t)"
assms: shows "LeftDerives1 \<alpha> i r \<gamma>"
CptnModEnv: fixes P :: "'b com option"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list"   and s :: "'b" assumes "(P, t) # xs \<in> cptn_mod" shows "(P, s) # (P, t) # xs \<in> cptn_mod"
elts_eq_empty_iff: fixes x :: "V" shows "(elts x = {}) = (x = []\<^sub>\<circ>)"
floor_divide_eq_div_numeral: fixes a :: "num"   and b :: "num" shows "\<lfloor>numeral a / numeral b\<rfloor> = numeral a div numeral b"
h2im: shows "h2 ` {0..1} \<subseteq> C"
valid_path_aux_Append: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "valid_path_aux cs as"   and "valid_path_aux (upd_cs cs as) as'" shows "valid_path_aux cs (as @ as')"
onorm_r: fixes r :: "real"   and f :: "'c \<Rightarrow>\<^sub>L 'd" assumes "0 < r" shows "norm f = (SUP x\<in>ball (0::'c) r. norm (f *\<^sub>v x)) / r"
bnd: fixes n :: "nat" shows "2 * \<bar>coeff f n\<bar> < m"
assms_13: shows "\<not> validate_ground_clause (int_clset S) (subst_cl (cl_ecl C) \<sigma>)"
faithful_functor_axioms: shows "faithful_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) F"
assms: fixes i :: "'c"   and j :: "'a" assumes "i \<in> CBasis"   and "j \<in> CBasis" shows "continuous F (\<lambda>x. cblinfun_apply (f x) j \<bullet>\<^sub>C i)"
prems_1: fixes p :: "'b"   and ls :: "'a list"   and eq :: "'a"   and gs :: "'a list" assumes "(p__, xs, ls__, eq__, gs__) = (p, [], ls, eq, gs)" shows "P"
rec_sigma_ex: fixes rf :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "primerec rf (Suc (length xs))" shows "(rec_exec (rec_sigma rf) (xs @ [x]) = 0) = (\<forall>t\<le>x. rec_exec rf (xs @ [t]) = 0)"
Sigma_triv: fixes a :: "'g"   and b :: "'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A \<and> b \<in> B a"
simps_3: shows "Abs None \<sqsubseteq> Abs None = True"
contents_def: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(self.contents()) = (\<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self then \<tau> \<Turnstile> true \<and> \<tau> \<Turnstile> res \<triangleq> if self.boss \<doteq> null then Set{self.salary} else self.boss.contents()->including\<^sub>S\<^sub>e\<^sub>t(self.salary) endif else \<tau> \<Turnstile> res \<triangleq> invalid)"
pred_mapping_mono_strong: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a option"   and R :: "'a \<Rightarrow> bool" assumes "pred_mapping Q P"   and "\<And>p. \<lbrakk>p \<in> dom P; Q (the (P p))\<rbrakk> \<Longrightarrow> R (the (P p))" shows "pred_mapping R P"
id1: shows "real_of_int (int (d ^ 2 ^ Suc k)) = real d ^ 2 ^ Suc k"
Ghyp: shows "\<forall>\<tau>. \<tau> \<in> U s \<and> \<tau> \<le> t \<longrightarrow> G (X \<tau>)"
cfun_map_below_ID: fixes e :: "'a \<rightarrow> 'a" assumes "e \<sqsubseteq> ID" shows "cfun_map\<cdot>e\<cdot>e \<sqsubseteq> ID"
coprime_h_c_poly: fixes c1 :: "'a mod_ring"   and c2 :: "'a mod_ring"   and h :: "'a mod_ring poly" assumes "c1 \<noteq> c2" shows "comm_monoid_mult_class.coprime (h - [:c1:]) (h - [:c2:])"
wt_elim_cases_16: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and C :: "qtname"   and sig :: "sig"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Methd C sig)\<Colon>T" obtains m :: "qtname \<times> methd" and T :: "ty" where "Ta__ = Inl T" and "is_class (prg E) C" and "methd (prg E) C sig = Some m" and "E,dt\<Turnstile>Body (declclass m) (stmt (mbody (mthd m)))\<Colon>-T"
node_ptr_kinds_eq2_h3: shows "|h3 \<turnstile> node_ptr_kinds_M|\<^sub>r = |h' \<turnstile> node_ptr_kinds_M|\<^sub>r"
borel_comp: fixes A :: "'a set" assumes "A \<in> sets borel" shows "- A \<in> sets borel"
ring_finite_field_ops_int: shows "ring_ops (finite_field_ops_int p) mod_ring_rel"
convinv_inverse: fixes c :: "'c \<Rightarrow>\<^sub>U 'd" shows "inv\<^sub>C (inv\<^sub>C c) = c"
nonzero': shows "(\<Sum>\<^sup>+ x. ennreal (pmf p x) * ennreal (measure_pmf.prob (f x) A)) \<noteq> 0"
c_manifold_axioms: shows "c_manifold (charts_submanifold W') k"
ccHeap_nonrec_def: fixes x :: "Vars.var"   and e :: "exp"   and exp :: "exp" shows "ccHeap_nonrec x e exp = (\<Lambda> n. CCfix_nonrec x e\<cdot>(Aexp_syn' n exp, ccExp_syn n exp))"
elims: fixes x :: "'c tree"   and y :: "nat" assumes "min_height x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'c tree" and uu_ :: "'c" and r :: "'c tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = min (min_height l) (min_height r) + 1"
prems: shows "set (coeffs (pCons a p)) \<subseteq> \<real>"
assms_3: shows "int (\<B> index ps) = \<Lambda>"
eq: shows "eq_upto_seq_inconsist P \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"
list_all3_conv_all_nth: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list" shows "list_all3 P xs ys zs = (length xs = length ys \<and> length ys = length zs \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i) (zs ! i)))"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b option"   and b :: "'c option" shows "rel_option R a b = ((a = None) = (b = None) \<and> (a \<noteq> None \<longrightarrow> b \<noteq> None \<longrightarrow> R (the a) (the b)))"
assert_bound_loop_code_def: shows "assert_bound_loop_code = AssertAllState''.assert_bound_loop assert_bound_code"
inv_inv: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "inv\<^bsub>C\<^esub> (inv\<^bsub>C\<^esub> x) = x"
homotopic_paths_imp_path: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "path p \<and> path q"
lm098: fixes Y :: "'a set"   and X :: "'a set" shows "runiq (X <|| Y)"
Sigma_empty_iff: fixes I :: "'b set"   and X :: "'b \<Rightarrow> 'c set" shows "(Sigma I X = {}) = (\<forall>i\<in>I. X i = {})"
assms: assumes "a \<noteq> (0::'a)" shows "set_integrable M A f"
normalize_content_ff_def: fixes p :: "'b fract poly" shows "normalize_content_ff p \<equiv> smult (inverse (content_ff p)) p"
meet_connection: fixes x :: "'c"   and y :: "'c" shows "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
apfst_convE: fixes q :: "'c \<times> 'd"   and f :: "'e \<Rightarrow> 'c"   and p :: "'e \<times> 'd" assumes "q = apfst f p" obtains x :: "'e" and y :: "'d" where "p = (x, y)" and "q = (f x, y)"
l_def: shows "l \<equiv> linepath (z + complex_of_real e2) (z + complex_of_real e1)"
fresh_string_set: fixes Xs :: "char list set"   and Y :: "char list" assumes "finite Xs" shows "fresh_string Xs Y \<notin> Xs"
tendsto_Im3: fixes g :: "'a \<Rightarrow> quat"   and a :: "quat"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im3 (g x)) \<longlongrightarrow> Im3 a) F"
span_sum: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and S :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> cspan S" shows "sum f A \<in> cspan S"
n_mult_closed: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "n_mult f \<in> carrier P"
splus_assoc: fixes s :: "'a"   and t :: "'a"   and v :: "'a" shows "local.punit.splus (s + t) v = local.punit.splus s (local.punit.splus t v)"
bind_assoc_pmf: fixes A :: "'d pmf"   and B :: "'d \<Rightarrow> 'c pmf"   and C :: "'c \<Rightarrow> 'b pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
Maclaurin_zero: fixes x :: "real"   and n :: "nat"   and diff :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "x = 0"   and "n \<noteq> 0" shows "(\<Sum>m<n. diff m (0::'a) / fact m * x ^ m) = diff 0 (0::'a)"
intro: fixes cl :: "'b potype" assumes "cl \<in> CompleteLattice" shows "CL_axioms cl"
FOREACHoci_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<Phi> :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'e set"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd nres"   and \<sigma>0 :: "'d" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. SPEC (\<lambda>xs. distinct xs \<and> S = set xs \<and> sorted_wrt R xs) \<bind> (\<lambda>xs. WHILE\<^sub>T\<^bsup>\<lambda>(it, \<sigma>). \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>\<^esup> (FOREACH_cond c) (FOREACH_body f) (xs, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
eq: shows "(cf ## \<omega>) !! N ## sdrop N \<omega> = sdrop N (cf ## \<omega>)"
domains_simp: fixes f :: "'a" assumes "arr f" shows "domains f = {local.dom f}"
nAct_not_same_active: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and n :: "nat"   and t :: "cnf llist"   and n' :: "enat" assumes "dynamic_component.nAct tCMP active c (enat n) t < dynamic_component.nAct tCMP active c n' t"   and "\<not> lfinite t \<or> n' - 1 < llength t" shows "\<exists>i\<ge>n. enat i < n' \<and> enat i < llength t \<and> active c (lnth t i)"
\<alpha>_\<rho>_l: fixes p :: "fm"   and i :: "int"   and bs :: "real list" assumes "iszlfm p (real_of_int i # bs)" shows "\<forall>(b, k)\<in>set (\<alpha>_\<rho> p). 0 < k \<and> numbound0 b \<and> isint b (real_of_int i # bs)"
not_outside_connected_component_lt: fixes S :: "'a set" assumes "bounded S"   and "2 \<le> DIM('a)" shows "- outside S = {x. \<forall>B. \<exists>y. B < norm y \<and> \<not> connected_component (- S) x y}"
Pair_eqD2: fixes x :: "'c"   and y :: "'d"   and x' :: "'c"   and y' :: "'d" assumes "(x, y) = (x', y')" shows "y = y'"
surj_to_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B" shows "surj_to f A (f ` A)"
isLim_def: fixes i :: "'b" shows "s.isLim i \<equiv> \<not> s.isSucc i"
perm_v_fm: fixes A :: "dbfm" assumes "supp A - {atom i} \<subseteq> atom ` Vs" shows "subst i' (Var sn') (ssubst (vquot_dbfm (insert i Vs) A) (insert i Vs) Fi) = ssubst (vquot_dbfm V' (subst_dbfm (DBVar sn) i A)) V' F'"
simps_2: fixes a :: "'c"   and x :: "'c list" shows "mset (a # x) = add_mset a (mset x)"
add_sheaf_spec_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "add_sheaf_spec U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.add_rel (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> (s \<pp>) (s' \<pp>)"
def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "basis_reduction_add_rows upw i fs = (if upw then basis_reduction_add_rows_loop i fs i else fs)"
hcomp_simps_1: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "hseq \<nu> \<mu>" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
a_very_costrict'': fixes x :: "'a" shows "(ad x = (1::'a)) = (x = (zero_class.zero::'a))"
act_emp_var: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c" assumes "unital_quantale_module act" shows "act x \<bottom> = \<bottom>"
xt1_9: fixes b :: "'a"   and a :: "'a"   and P :: "bool" assumes "b \<in> U"   and "a \<in> U"   and "b <\<^sub>o\<^sub>w a"   and "a <\<^sub>o\<^sub>w b" shows "P"
fmadd_disjnt: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" assumes "fdisjnt (fmdom m) (fmdom n)" shows "m ++\<^sub>f n = n ++\<^sub>f m"
closedin_Int_closure_of: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "closedin (subtopology X S) T = (S \<inter> X closure_of T = T)"
indRelR_cond_respection_iff_indRelL_cond_respection: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_pred Rel Pred) = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_respects_pred Rel Pred)"
simps_2: fixes z :: "gauss_int" shows "ImZ (gauss_cnj z) = - ImZ z"
section_def: fixes f :: "'b" shows "D.section f \<equiv> \<exists>g. D.ide (g \<cdot>\<^sub>D f)"
A_2: shows "\<forall>x\<in>X. I x = I' x"
U_2: shows "{u \<in> V. u x \<le> real_of_int m} = \<Union> U"
vsubset_vunion_leftD_1: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<union>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> C"
f: shows "f \<in> borel_measurable (lebesgue_on (g ` S))"
set_iterator_image_foldli_conv: fixes g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "set_iterator_image g (foldli xs) = foldli (map g xs)"
simps_28: fixes x1 :: "'b" shows "set_trm (Var x1) = {x1}"
inf_left_idem: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf x y) = inf x y"
extending_Increasing: fixes F :: "'a program"   and func :: "'a \<Rightarrow> 'e" shows "extending (\<lambda>G. reachable (extend h F \<squnion> G)) h F (Increasing (func \<circ> f)) (Increasing func)"
is_path_undir_simps_2: fixes v :: "'v"   and w :: "'w"   and v' :: "'v" shows "is_path_undir G v [(v, w, v')] v' = ((v, w, v') \<in> E \<or> (v', w, v) \<in> E)"
that: fixes q :: "'c" assumes "d_reaches M (initial M) v q" shows "thesis"
terminationPseudo_def: fixes t :: "nat"   and c :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "terminationPseudo t c Q \<equiv> initReachable c \<and> card Proc \<le> card Q + t \<longrightarrow> (\<exists>c'. qReachable c Q c' \<and> (\<exists>v. initReachable c' \<and> <\<bottom>, outM v> \<in># msgs c'))"
simps_20: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (Minus e1 e2) (Plus v va) = False"
assms_3: shows "\<forall>x\<in>fv t - X. \<exists>a. \<Gamma> (Var x) = Var a"
antisym: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
WL_def: fixes p :: "'a process_name" shows "WL p = (\<lambda>s. W (s p) \<union> ghost_honorary_grey (s p))"
setops\<^sub>s\<^sub>s\<^sub>t_wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_2: fixes A :: "('fun, 'a) stateful_strand_step list" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t A)"
HcarrierNH: shows "H \<subseteq> carrier (G\<lparr>carrier := N <#> H\<rparr>)"
valid_sops_stmt_sb: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b"   and s' :: "stmt"   and t' :: "nat" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)"   and "(s', t') \<in> prog_configs sb" shows "valid_sops_stmt t' s'"
lexordp_antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.order less_eq less"   and "ord.lexordp less xs ys"   and "ord.lexordp less ys xs" shows "False"
simps: fixes p :: "real poly"   and q :: "real poly" shows "smods p q = (if p = 0 then [] else p # smods q (- (p mod q)))"
elims_2: fixes x :: "'b aexp" assumes "is_lit x" obtains uu_ :: "value" where "x = L uu_"
hyps_2: shows "\<forall>c\<in>S. \<exists>Ss. K c = Some Ss \<and> list_all2 (\<lambda>x1 x2. has_sort (sub, tcs) x1 x2 \<and> (True \<and> True \<longrightarrow> (\<forall>x. sort_leq sub x2 x \<longrightarrow> wf_osig (sub, tcs) \<longrightarrow> has_sort (sub, tcs) x1 x))) Ts Ss"
assms_1: fixes k :: "nat" assumes "j < i"   and "k \<le> j" shows "f1 ! k = f2 ! k"
HEndPhase1_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "Inv2b s" shows "HInv4c s' p"
simps_7: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 M\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = (if \<psi>\<^sub>1 M\<^sub>n \<psi>\<^sub>2 \<in> X then \<psi>\<^sub>1[X]\<^sub>\<nu> R\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu> else false\<^sub>n)"
valid_eval_muaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'muaux"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and nt :: "nat"   and res :: "event_data option list set list"   and aux' :: "'muaux"   and res' :: "event_data option list set list"   and auxlist' :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "valid_muaux args cur aux auxlist"   and "cur \<le> nt"   and "eval_muaux args nt aux = (res, aux')"   and "eval_until (args_ivl args) nt auxlist = (res', auxlist')" shows "res = res' \<and> valid_muaux args cur aux' auxlist'"
distinct_1: fixes name1' :: "name"   and name2' :: "name"   and pi' :: "pi" shows "\<zero> \<noteq> name1'{name2'}.pi'"
adj_sub_geq2_fact: fixes i :: "'d"   and x2 :: "('d, 'e) trm"   and j :: "'f + 'f"   and \<sigma> :: "('d, 'g, 'f, 'h) subst_scheme"   and x1 :: "('d, 'i) trm" assumes "i \<in> SIGT x2"   and "j \<in> (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Inl i \<in> ({Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}) \<inter> {Inl x |x. x \<in> SIGT x1 \<or> x \<in> SIGT x2}"
mem_UnitsD: fixes u :: "'a" assumes "u \<in> additive.Units" shows "additive.invertible u \<and> u \<in> R"
div_0: 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 a :: "'b" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
pos_mod_bound: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) < b" shows "a mod b < b"
chamber_system_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplex.chamber_system X \<equiv> {C. SimplicialComplex.maxsimp X C}"
ValFF_below_iff: fixes x :: "ValD" shows "(ValFF \<sqsubseteq> x) = (x = ValFF)"