hcju/isabelleps huggingface dataset:
frequently_eventually_at_top: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "frequently P at_top"   and "eventually Q at_top" shows "\<exists>\<^sub>F x in at_top. P x \<and> (\<forall>y\<ge>x. Q y)"
reducecoeffh_numbound0: fixes t :: "MIR.num"   and g :: "int" assumes "numbound0 t" shows "numbound0 (reducecoeffh t g)"
assert_all_sat: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('d \<times> 'e atom) list"   and v :: "(nat, 'e) mapping" assumes "\<triangle> t"   and "assert_all t as = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>t t \<and> \<langle>v\<rangle> \<Turnstile>\<^sub>a\<^sub>s (snd ` set as)"
pg_3: shows "pathfinish pg = p + complex_of_real (h p)"
ccompatible: fixes cc :: "('d, real) cconstraint"   and k :: "'d \<Rightarrow> nat"   and X :: "'d set" assumes "\<forall>(x, m)\<in>collect_clock_pairs cc. m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<nat>" shows "ccompatible {region X I r |I r. valid_region X k I r} cc"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>v. (\<not> odd (Suc v) \<Longrightarrow> P (Suc v div 2)) \<Longrightarrow> P (Suc v)" shows "P a0"
set_mset_diff: fixes M :: "'a multiset"   and A :: "'a set" shows "set_mset (M -s A) = set_mset M - A"
WT_gpv_ContD: fixes \<Gamma> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and out :: "'h"   and c :: "('j, 'h, 'i) rpv"   and input :: "'i" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
nat_of: fixes x :: "'b sat" shows "nat_of x \<in> {..LENGTH('b)}"
insert_dom_eq: fixes f :: "'a \<Rightarrow> 'b option"   and f' :: "'a \<Rightarrow> 'c option"   and l :: "'a"   and x :: "'b"   and x' :: "'c" assumes "dom f = dom f'" shows "dom (f(l \<mapsto> x)) = dom (f'(l \<mapsto> x'))"
Int_Diff: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<inter> B - C = A \<inter> (B - C)"
subset_Un_eq: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subseteq> B) = (A \<union> B = B)"
Derives1_count_terminals_leq: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "count_terminals a \<le> count_terminals b"
wf_update: fixes i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "i \<in> agents"   and "total_preorder_on alts Ri'" shows "pref_profile_wf agents alts (R(i := Ri', i := Ri'))"
sp'_4: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and A1 :: "'a"   and A2 :: "'a"   and A3 :: "'a"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "fb_impossibility_3_3 agents alts scf A1 A2 A3 a b c" shows "a \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<and> (b \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<and> b \<notin> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<or> c \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<and> c \<notin> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c)) \<or> b \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<and> c \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<and> b \<notin> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<or> a \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<and> (b \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<or> c \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c)) \<or> c \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<and> c \<notin> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<and> b \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<or> (c \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<longrightarrow> b \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<longrightarrow> b \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c)) \<and> (c \<in> scf (fb_impossibility_3_3.R2 A1 A2 A3 a b c) \<longrightarrow> c \<in> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c) \<or> b \<notin> scf (fb_impossibility_3_3.R10 A1 A2 A3 a b c))"
bij_convert: shows "bij_betw convert_perm {\<sigma>. \<sigma> permutes {..<n}} {\<sigma>. \<sigma> permutes Roots}"
elims: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and x :: "'c"   and xa :: "('a \<times> 'b) list \<times> 'c"   and y :: "('a \<times> 'b) list" assumes "oalist_raw.sort_oalist_aux rep_key_order x xa = y" obtains ko :: "'c" and xs :: "('a \<times> 'b) list" and ox :: "'c" where "x = ko" and "xa = (xs, ox)" and "y = (if ko = ox then xs else comparator.sort_oalist (key_compare (rep_key_order ko)) xs)"
member_remove: fixes x :: "'e"   and y :: "'e"   and A :: "'e set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"
assms: shows "QD('d\<^sub>1) = QD('d\<^sub>2)"
arr_cod: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.arr (V.cod f)"
sset_sdrop: fixes n :: "nat"   and xs :: "'b stream" shows "sset (sdrop n xs) \<subseteq> sset xs"
rel_compp: fixes R :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_bset (R OO S) = rel_bset R OO rel_bset S"
PP_PPf_eqv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (eqv (PP \<langle>\<phi>\<rangle>) (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)))"
real_lim_then_eventually_real: fixes u :: "'a \<Rightarrow> ereal"   and l :: "real"   and F :: "'a filter" assumes "(u \<longlongrightarrow> ereal l) F" shows "\<forall>\<^sub>F n in F. u n = ereal (real_of_ereal (u n))"
comp_fun_commute_relcomp_fold: fixes S :: "('b \<times> 'c) set" assumes "finite S" shows "comp_fun_commute (\<lambda>(x, y) A. Finite_Set.fold (\<lambda>(w, z) A'. if y = w then insert (x, z) A' else A') A S)"
Rep_uint32'_def: shows "Rep_uint32' = Rep_uint32"
setlistmapim_comp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and xs :: "'c set list" shows "(f \<circ> g) \<Turnstile> xs = f \<Turnstile> (g \<Turnstile> xs)"
d_def: fixes x :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x = (1::'a) \<sqinter> x \<cdot> \<top>"
lists_fc_succ_def: fixes xss :: "('b \<times> 'b) list list" shows "lists_fc_succ xss = (let sxss = \<Union> (sset xss) in (\<lambda>x. lists_succ xss (swap_in sxss x)))"
paths_hits_winning_strategy_is_winning: fixes v :: "'a" assumes "winning_strategy p \<sigma>' v"   and "v \<in> lset P'" shows "winning_path p P'"
1: shows "\<forall>m<n. \<forall>x xa. (num_params (tm_poly xa) \<le> length I \<longrightarrow> develops_at_within e a I \<longrightarrow> x \<in>\<^sub>i range_tm e xa \<longrightarrow> x ^ m \<in>\<^sub>i range_tm e (tm_pow prec ord I a xa m)) \<and> num_params (tm_poly (tm_pow prec ord I a xa m)) \<le> num_params (tm_poly xa)"
If_cases: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool"   and h :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "finite A" shows "(\<Sum>x\<in>A. if P x then h x else g x) = sum h (A \<inter> {x. P x}) + sum g (A \<inter> - {x. P x})"
det_I: shows "det (mat (1::'b)) = (1::'b)"
IP: fixes i :: "nat" shows "(f__ i, f__ (Suc i)) \<in> rwof_rel init cond step"
eint_2_minus_1_mult: fixes b :: "eint" shows "eint 2 * b - b = b"
nonneg_h: fixes n :: "nat"   and x :: "(real, 'n) vec" shows "0 \<le> h n x"
limsup_infy: shows "limsup (\<lambda>x. ereal (aa (x + 1) / prod aa {0..x} powr (2 + \<delta>) * (1 / bb (x + 1)))) = \<infinity>"
inDomConc: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "x \<notin> dom (C a)"   and "x \<notin> dom (C (list2FWpolicy p))" shows "x \<notin> dom (C (list2FWpolicy (a # p)))"
mk_eqcl_fixpt_trans: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and p :: "nat"   and bs :: "bool list" assumes "wf_dfa M v"   and "dfa_is_node M p"   and "is_alph v bs" shows "dfa_trans (min_dfa M) (fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! p) bs = fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_trans M p bs"
qSwap_ident: fixes zs :: "'c"   and x :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "X #[[x \<and> x]]_zs = X"
composition': fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b"   and \<Psi>''' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "(\<Psi> \<otimes> \<Psi>'') \<otimes> \<Psi>''' \<simeq> (\<Psi>' \<otimes> \<Psi>'') \<otimes> \<Psi>'''"
P1: shows "\<not> Col A B C \<and> Coplanar A B C X"
crename_ImpL: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name" assumes "R[a\<turnstile>c>b] = ImpL <c>.M y.N z"   and "c \<sharp> (a, b, N, R)"   and "y \<sharp> (z, M, R)" shows "\<exists>M' N'. R = ImpL <c>.M' y.N' z \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> c \<sharp> N' \<and> y \<sharp> M'"
hRe_hcomplex_of_complex: fixes z :: "complex" shows "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
exists_Ord_Least_iff: fixes P :: "V \<Rightarrow> bool" shows "(\<exists>\<alpha>. Ord \<alpha> \<and> P \<alpha>) = (\<exists>\<alpha>. Ord \<alpha> \<and> P \<alpha> \<and> (\<forall>\<beta><\<alpha>. Ord \<beta> \<longrightarrow> \<not> P \<beta>))"
Gromov_product_le_dist_1: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "Gromov_product_at e x y \<le> dist e x"
simps_1: fixes x :: "real"   and y :: "real" shows "(ereal x < ereal y) = (x < y)"
cut_less_inext_nth_card_eq2: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "card I \<le> Suc n" shows "card (I \<down>< (I \<rightarrow> n)) = card I - Suc 0"
arcosh_real_at_top: shows "filterlim arcosh at_top at_top"
segment_sub: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D" shows "segment D a \<subseteq> carrier D"
mgu_complete: fixes AAA :: "'a set set"   and \<sigma> :: "'s" assumes "finite AAA"   and "\<forall>AA\<in>AAA. finite AA"   and "is_unifiers \<sigma> AAA" shows "\<exists>\<tau>. mgu AAA = Some \<tau>"
bl_to_bin_lt2p: fixes bs :: "bool list" shows "bl_to_bin bs < 2 ^ length bs"
2: shows "fmdom' (fmupd v x (fmrestrict_set X s)) = insert v X"
bex1_def: fixes X :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(\<exists>!x. x \<in> X \<and> P x) = ((\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y))"
span_subsetI: fixes ws :: "complex vec set"   and us :: "complex vec set" assumes "ws \<subseteq> carrier_vec d"   and "us \<subseteq> local.span ws" shows "local.span us \<subseteq> local.span ws"
checkpoint_s_def: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('c, 'd) state" assumes "Incremental_State_Ops_Simplex assert_bound init check" shows "Incremental_State_Ops_Simplex.checkpoint_s s = \<B>\<^sub>i s"
snd_normalize_quot_nonzero: fixes x :: "'b \<times> 'b" shows "snd (normalize_quot x) \<noteq> (0::'b)"
nodes_of_uses'_correct: fixes g :: "'g"   and nodes_of_uses :: "('val, 'node set) mapping"   and "next" :: "'node \<times> 'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_uses (uninst_code.ssa.useNodes_of (u_g g s1_uses) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)"   and "uninst_code.ssa.trivial (u_g g s1_uses) (p_g g s1_phis) g (snd next)" shows "uninst_code.phi_equiv_mapping (p_g g s1_phis) g (uninst_code.nodes_of_uses' g next (uninst_code.substitution_code (p_g g s1_phis) g next) (Mapping.keys (uninst_code.ssa.phidefNodes (p_g g s1_phis) g)) nodes_of_uses) (uninst_code.ssa.useNodes_of (\<lambda>_. uninst_code.uses'_codem (u_g g s1_uses) g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_uses) g)"
xBij: shows "conjx \<in> Bij (subgroups_of_size p)"
nn_integral_cong_simp: fixes M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal"   and v :: "'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M =simp=> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
exist_comb: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "\<exists>z. heap_sep_algebra.tern_rel plus sep_disj x y z"
col2__eq: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p" assumes "Col A X Y"   and "Col B X Y"   and "\<not> Col A X B" shows "X = Y"
x: shows "x = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i1"
If2_def: fixes Q :: "tr"   and x :: "'c"   and y :: "'c" shows "If2 Q x y = If Q then x else y"
prj_simps_13: shows "C.dom \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1 = HH\<mu>\<nu>H\<pi>\<rho>.chine"
enum_works: fixes N :: "nat set" assumes "infinite N" shows "N = range (enum N) \<and> strict_mono (enum N)"
simps_2: shows "cospan_shape.Dom cospan_shape.BB = cospan_shape.BB"
monom_mult_uminus_left: fixes c :: "'d"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "MPoly_Type_Class.punit.monom_mult (- c) t p = - MPoly_Type_Class.punit.monom_mult c t p"
deg_g_nzero: shows "deg R (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<noteq> 0"
B_simps_4: fixes h :: "nat" shows "(N0 \<in> B h) = (h = 0)"
subsumes_trans: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset" assumes "subsumes C D"   and "subsumes D E" shows "subsumes C E"
of_int_round_le: fixes x :: "'a" shows "of_int (round x) \<le> x + (1::'a) / (2::'a)"
bind_eq_return_spmf: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf"   and x :: "'a" shows "(p \<bind> f = return_spmf x) = ((\<forall>y\<in>set_spmf p. f y = return_spmf x) \<and> lossless_spmf p)"
ev_suntil: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "(\<phi> suntil \<psi>) \<omega>" shows "ev \<psi> \<omega>"
non_emptyE: fixes A :: "'c set" assumes "A \<noteq> {}" obtains x :: "'c" where "x \<in> A"
X_2: assumes "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map permutations xss)" shows "distincts yss"
filter_empty_conv: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
n_val_representative: fixes P :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds" shows "\<nu>\<^bsub>K P\<^esub> \<in> P"
tailmarked_def: shows "tailmarked = (\<lambda>(x, y). outmarked x)"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a atom"   and b :: "'b atom" assumes "rel_atom R a b" obtains x1 :: "predicate" and x2 :: "'a list" and y1 :: "predicate" and y2 :: "'b list" where "a = predAtm x1 x2" and "b = predAtm y1 y2" and "x1 = y1" and "list_all2 R x2 y2"   | x1a :: "'a" and x2a :: "'a" and y1a :: "'b" and y2a :: "'b" where "a = Eq x1a x2a" and "b = Eq y1a y2a" and "R x1a y1a" and "R x2a y2a"
next_subset_dfs: fixes succs :: "'a \<Rightarrow> 'a list"   and is_node :: "'a \<Rightarrow> bool"   and invariant :: "'b \<Rightarrow> bool"   and ins :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and memb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and empt :: "'b"   and S :: "'b"   and xs :: "'a list"   and x :: "'a" assumes "DFS succs is_node invariant ins memb empt"   and "invariant S"   and "list_all is_node xs"   and "x \<in> set xs" shows "memb x (gen_dfs succs ins memb S xs)"
predtrans_empty: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "mono \<tau>" shows "\<forall>i. (\<tau> ^^ i) {} \<subseteq> (\<tau> ^^ (i + 1)) {}"
couSim: shows "weak_barbed_coupled_simulation (indRelRTPO TRel) (STCalWB SWB TWB)"
l1_step2_refines_a0i_running_skip_i: fixes A :: "agent"   and B :: "agent"   and gnx :: "msg"   and Rb :: "fid_t" shows "{R01iai} (a0i_running A B (Exp gnx (NonceF (Rb $ ny))))\<^sup>=, l1_step2 Rb A B gnx {> R01iai}"
solves_ode_def: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set" shows "(y solves_ode f) T X = ((y has_vderiv_on (\<lambda>t. f t (y t))) T \<and> y \<in> T \<rightarrow> X)"
p'': shows "path \<Gamma> (x, y) (zip (y # p) p) (last (x # butlast (y # p)), sink \<Delta>)"
mydivcode1: fixes k :: "nat" shows "mydiv 0 k = 0"
pseudo_proj_True_preimage: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_True n w" shows "pseudo_proj_True n -` {w} = (\<Inter>i\<in>{m. Suc m \<le> n}. (\<lambda>w. w !! i) -` {w !! i})"
summable: shows "(\<lambda>p. complex_of_real (ln (real p)) / (of_nat p powr s - 1) + B p s) abs_summable_on {p. prime p}"
sum_fun_image_sup: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and z :: "'c" assumes "finite A" shows "(\<Sum> (f ` A) \<le> z) = (\<forall>a\<in>A. f a \<le> z)"
OpCatCat: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "Category C" shows "Category (Op C)"
lookup_nil: fixes e :: "('f, 'g, 'h) env" shows "lookup e [] = Some e"
prefix_same_cases: fixes xs\<^sub>1 :: "'b list"   and ys :: "'b list"   and xs\<^sub>2 :: "'b list" assumes "xs\<^sub>1 \<le>p ys"   and "xs\<^sub>2 \<le>p ys" shows "xs\<^sub>1 \<le>p xs\<^sub>2 \<or> xs\<^sub>2 \<le>p xs\<^sub>1"
completenessForUNSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "\<not> satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "formulaFalse F0 (elements (getM state)) \<and> decisions (getM state) = []"
simps: fixes T :: "(nat \<times> linear_poly) list"   and T_old :: "(nat \<times> linear_poly) list"   and BIL :: "(nat, 'a \<times> 'b) mapping"   and BIU :: "(nat, 'a \<times> 'b) mapping"   and V :: "(nat, 'b) mapping"   and U :: "bool"   and UC :: "'a list option" shows "\<T>_update T (State T_old BIL BIU V U UC) = State T BIL BIU V U UC"
jumpNestingOk_stmt_simp: fixes jmps :: "jump set"   and s :: "stmt" shows "jumpNestingOk jmps (In1r s) = jumpNestingOkS jmps s"
u_def: shows "u = (\<lambda>n. SOME z. z \<in> A n \<and> z < x)"
RUV_sound: fixes u :: "'a"   and w :: "('b, 'c) label list el_step list"   and s' :: "'a list"   and c' :: "'a list multiset"   and U :: "'a set"   and V :: "'a set" assumes "(([u], {#}), w, s', c') \<in> trcl (ntrp fg)"   and "atUV U V ({#s'#} + c')" shows "\<exists>Ml Me. (u, Ml, Me) \<in> RUV_cs fg U V \<and> Ml \<subseteq> mon_loc fg w \<and> Me \<subseteq> mon_env fg w"
lh_drv_prem: shows "\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1"
count_empty: fixes a :: "'k" shows "count {#} a = 0"
none2: fixes a :: "'a" assumes "x = Some a"   and "y = None" shows "thesis"
tau_converg: fixes x :: "nat"   and i :: "nat" assumes "determined i x" shows "\<tau> i x \<down>= e_nth (block i (the (dettime i x))) x"
prepare_mainpart_lemma: fixes args :: "nat list"   and m :: "nat" assumes "args \<noteq> []" shows "\<exists>stp ln rn. steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main) stp = (0, Bk # Oc \<up> Suc m, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> bl_bin (<args>) @ Bk \<up> rn)"
val_1mx: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 \<le> v (1\<^sub>r \<plusminus> -\<^sub>a x)" shows "0 \<le> v x"
ground_l\<^sub>2: shows "ground\<^sub>l l"
Tp: shows "T\<^sub>p (Lxy init {x, y}) (Lxy qs {x, y}) Strat2 + length sws = ALGxy_det Strat qs init x y + ALG_Pxy Strat qs init x y"
zero_set_def: shows "zero_set = (\<lambda>f. {x. f x = 0})"
power2_eq_imp_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times x 2 = power.power one times y 2"   and "less_eq zero x"   and "less_eq zero y" shows "x = y"
J_rad_unit: fixes R :: "('h, 'i) Ring_scheme"   and x :: "'h" assumes "Ring R"   and "\<not> zeroring R"   and "x \<in> J_rad R" shows "\<forall>y. y \<in> carrier R \<longrightarrow> Unit R (1\<^sub>r\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> (-\<^sub>a\<^bsub>R\<^esub> x) \<cdot>\<^sub>r\<^bsub>R\<^esub> y)"
PRE: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "push_effect_locale c s t f l u v" shows "Network.push_precond c f l (u, v)"
id_oracle_def: fixes s :: "'b"   and x :: "'c" shows "1\<^sub>I s x = Pause x (\<lambda>x. Done (x, s))"
pt_sol: shows "0 < t \<and> t < p \<and> p * t = sum4sq_int (a, b, c, d)"
\<tau>_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and t :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.\<tau> \<delta> t = t - \<delta>"
assms: fixes kv :: "'b \<times> 'c" assumes "kv \<in> set xs" shows "f kv = g kv"
snoc_1: fixes vs :: "v list"   and tvs :: "t list" assumes "ves = $$* vs"   and "\<S>\<bullet>\<C> \<turnstile> ves : [] _> tvs" shows "tvs = map typeof vs"
left_associated_product: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "\<exists>w. P w x \<and> P w y \<and> P w z" shows "\<forall>w. P w ((x \<otimes> y) \<otimes> z) = ((P w x \<and> P w y) \<and> P w z)"
lt_set_iff: fixes t :: "'t"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "(t \<in> lt_set F) = (\<exists>f\<in>F. f \<noteq> 0 \<and> lt f = t)"
isCont_sgn: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "isCont f a"   and "f a \<noteq> (0::'b)" shows "isCont (\<lambda>x. sgn (f x)) a"
hom_dvd: fixes hom :: "'a \<Rightarrow> 'b"   and p :: "'a"   and q :: "'a" assumes "comm_monoid_mult_hom hom"   and "p dvd q" shows "hom p dvd hom q"
tmod2_4: shows "[mn_] \<notin> defined_mods s \<and> no_dup_types ds_ \<and> evaluate_decs ck [mn_] env s ds_ (s2_, Rerr err_)"
IH: shows "m = s \<lhd>/ m ++\<^sub>f s \<lhd> m"
multi_count_eq: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(\<forall>a. count M a = count N a) = (M = N)"
Max_lit_eq_pos_or_neg_Max_atm: fixes C :: "'b literal multiset" assumes "C \<noteq> {#}" shows "Max_mset C = Pos (Max (atms_of C)) \<or> Max_mset C = Neg (Max (atms_of C))"
ih4: shows "pi \<bullet> (\<parallel><B>\<parallel>) = \<parallel><B>\<parallel>"
rel_interior_sum: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "rel_interior (S + T) = rel_interior S + rel_interior T"
matrix_inv_matrix_mul: fixes A :: "(('d, 'e) vec, 'f) vec"   and B :: "(('d, 'g) vec, 'e) vec" assumes "invertible A"   and "invertible B" shows "(A ** B)\<^sup>-\<^sup>1 = B\<^sup>-\<^sup>1 ** A\<^sup>-\<^sup>1"
seq_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "c\<^sub>1 \<noteq> Stop" shows "\<exists>c\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem'\<rangle> \<and> c' = c\<^sub>1' ;; c\<^sub>2"
inheritable_in_def: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and pack :: "pname" shows "G \<turnstile> membr inheritable_in pack = (case accmodi membr of Private \<Rightarrow> False | Package \<Rightarrow> pid (declclass membr) = pack | _ \<Rightarrow> True)"