hcju/isabelleps huggingface dataset:
positively_responsive_prefer: fixes P :: "'i \<Rightarrow> ('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "profile A Is P"   and "hasw [x, y] A"   and "card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x} < card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y}" shows "x \<^bsub>(scf P)\<^esub>\<prec> y"
P'_witness: shows "\<forall>X\<in>P'. \<exists>x. x \<in> X \<inter> C"
le_measure_iff: fixes M :: "'a measure"   and N :: "'a measure" shows "(M \<le> N) = (if space M = space N then if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"
axioms_1: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partial_equivalence A equiv" shows "symmetric A equiv"
assms_2: shows "\<I>2, \<I>3 \<turnstile>\<^sub>C CNV callee s \<surd>"
sel_3: fixes z :: "quat" shows "Im2 (Quaternions.cnj z) = - Im2 z"
h8: shows "[(ainv * b')\<^sup>2 + 1 = 0] (mod 4 * k + 3)"
simps_6: fixes I :: "('b, 'c, 'a) interp"   and t :: "('b, 'a) trm" shows "dterm_sem I (Differential t) = directional_derivative I t"
pairself_def: fixes f :: "'c \<Rightarrow> 'd" shows "pairself f \<equiv> \<lambda>x. case x of (u, v) \<Rightarrow> (f u, f v)"
intros: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c'"   and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'"   and "stack c' = stack c" shows "c \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> c'"
no: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
Nil: fixes Env :: "env"   and B :: "lname set" shows "Env\<turnstile> B \<guillemotright>\<langle>[]\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
that: fixes vs :: "('a \<times> 'b) list"   and xs :: "('a \<times> 'b) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs" shows "thesis"
null_sets: shows "null_sets (distr M N X) = null_sets N"
update_def: fixes s :: "state"   and vn :: "vname"   and v :: "nat" shows "s[vn::=v] \<equiv> case vn of Glb gn \<Rightarrow> case s of st g l \<Rightarrow> st (g(gn := v)) l | Loc ln \<Rightarrow> case s of st g l \<Rightarrow> st g (l(ln := v))"
prems: fixes x :: "nat" assumes "x < count_weights False (remove_weights (Pool m1 m2))" shows "f x = extract_weights False (Pool m1 m2) x"
fresh_not_KeyWithNonce: fixes K :: "nat"   and evs :: "event list"   and NB :: "nat" assumes "Key K \<notin> used evs" shows "\<not> KeyWithNonce K NB evs"
wtp: shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
rquot_succs_def: fixes M :: "nat bdd list \<times> bool list" shows "rquot_succs M = (\<lambda>n x. [bdd_lookup (fst M ! x) (replicate n False)])"
assms_7: fixes x :: "'a" assumes "P x" shows "\<alpha> (f x) = g (\<alpha> x)"
meet_related2: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x" shows "x \<sqinter> y = y"
Join_equality: fixes A :: "'b set"   and sup :: "'b" assumes "is_Sup A sup" shows "\<Squnion>A = sup"
below_trans: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.po below"   and "below x y"   and "below y z" shows "below x z"
size_multiset_union: fixes f :: "'a \<Rightarrow> nat"   and M :: "'a multiset"   and N :: "'a multiset" shows "size_multiset f (M + N) = size_multiset f M + size_multiset f N"
mset_right_cancel_union: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" assumes "a \<in># A + B"   and "a \<notin># B" shows "a \<in># A"
path_def: fixes x :: "'obj"   and y :: "'obj"   and p :: "'arr list" shows "path x y p \<equiv> p = [] \<and> x = y \<and> x \<in> Obj \<or> p \<noteq> [] \<and> x = Dom (hd p) \<and> y = Cod (last p) \<and> (\<forall>n. 0 \<le> n \<and> n < length p \<longrightarrow> p ! n \<in> Arr) \<and> (\<forall>n. 0 \<le> n \<and> n < length p - 1 \<longrightarrow> Cod (p ! n) = Dom (p ! (n + 1)))"
elims: fixes x :: "linear_poly"   and y :: "nat" assumes "dim_poly x = y" obtains p :: "linear_poly" where "x = p" and "y = (if vars p = {} then 0 else Max (vars p) + 1)"
siso0_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "siso0 c"   and "mustT c s"   and "(c, s) \<rightarrow>c (c', s')" shows "siso0 c'"
hom_chain_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and p :: "nat" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T" shows "chain_map p f \<in> hom (relcycle_group p X S) (relcycle_group p Y T)"
open_fpxs_def: fixes U :: "'a fpxs set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
conj_disj_distrib2: fixes y :: "'b"   and z :: "'b"   and x :: "'b" shows "(y OR z) AND x = y AND x OR z AND x"
map_poly_pCons_0_as_mult: fixes p :: "'c poly poly" shows "map_poly (pCons (0::'c)) p = [:pCons (0::'c) 1:] * p"
assms_4: shows "t mod cycleLength (nC k) mem schedule (nC k)"
mset_link: fixes h1 :: "'b hp"   and h2 :: "'b hp" shows "mset_hp (link h1 h2) = mset_hp h1 + mset_hp h2"
simps_3: fixes x :: "real"   and y :: "real" shows "(real_min_top.R x < real_min_top.R y) = (x < y)"
Suc_1: fixes B :: "nat set" assumes "n = card B"   and "finite A"   and "B \<subseteq> A"   and "\<forall>b\<in>B. f b \<noteq> 0" shows "prod f (A - B) = prod f A / prod f B"
Int_atLeastAtMost: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word"   and d :: "'a word" shows "signed.atLeastAtMost a b \<inter> signed.atLeastAtMost c d = signed.atLeastAtMost (signed.max a c) (signed.min b d)"
2: shows "LeftDerives1 (ladder_\<alpha> \<alpha> D L 1) (ladder_i L 1) rule (a1 @ snd rule @ a2)"
tab_simps_2: shows "src f.tab = src (tab\<^sub>0 f)"
finite_INT: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "\<exists>x\<in>I. finite (A x)" shows "finite (\<Inter> (A ` I))"
less_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "(x < y) = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
rt_fresher_update_other: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "dip \<in> kD rt"   and "dip \<noteq> ip" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> update rt ip r"
prf_domain_length: fixes x :: "bool list" assumes "x \<in> prf_domain" shows "length x = prf_dlen"
assms_2: shows "M 0 i + M i 0 = (0::'a)"
in_on_out: fixes H :: "ocircline"   and z :: "complex_homo" shows "in_ocircline H z \<or> on_ocircline H z \<or> out_ocircline H z"
lrev_work1_lrev_work_eq: shows "lrev_work1 = lrev_work"
prod_c0: shows "\<forall>A\<in>set As. production N (CA_of A) = {A}"
dim_extract_matrix_1: fixes as :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "dim_row (extract_matrix as m n) = m"
lookup_pair_eq_0: fixes xs :: "('b \<times> 'c) list"   and k :: "'b" assumes "tc.oalist_inv_raw xs" shows "(lookup_pair_tc xs k = (0::'c)) = (k \<notin> fst ` set xs)"
assms: shows "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = None"
compet_lb0: fixes Aoff :: "'b list \<Rightarrow> 'b list \<Rightarrow> (nat \<times> nat list) list"   and cruel :: "nat \<Rightarrow> 'b list"   and Aon :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and c :: "real"   and S0 :: "'b list set"   and s0 :: "'b list"   and l :: "real"   and a :: "real" assumes "\<And>rs s0. length (Aoff s0 rs) = length rs"   and "\<And>n. cruel n \<noteq> []"   and "compet Aon c S0"   and "0 \<le> c"   and "s0 \<in> S0"   and "\<forall>\<^sub>F n in sequentially. l \<le> real (T_on Aon s0 (cruel n)) / (real (T_off Aoff s0 (cruel n)) + a)"   and "LIM n sequentially. real (T_off Aoff s0 (cruel n)) :> at_top"   and "0 < l"   and "\<And>n. static s0 (cruel n)" shows "l \<le> c"
not_know_k_not_ine: fixes A :: "specID"   and m :: "Keys" assumes "\<not> know A (kKS m)" shows "\<not> ine A (kE m)"
take_Suc_j_b_decompose: shows "take (Suc j) b = take j b @ [a ! i]"
stake_invert_Nil: fixes n :: "nat"   and s :: "'c stream" shows "(stake n s = []) = (n = 0)"
rel_interior_injective_on_span_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "bounded_linear f"   and "inj_on f (span S)" shows "rel_interior (f ` S) = f ` rel_interior S"
assms_3: fixes s :: "'a set" assumes "s \<in> simplices" shows "card s = n + 2"
random_bst_altdef: fixes A :: "'b set" assumes "finite A" shows "random_bst A = map_pmf bst_of_list (pmf_of_set (permutations_of_set A))"
aboveS_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
rel_map_1: fixes Sb :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and i :: "'i \<Rightarrow> 'g"   and x :: "'i list"   and y :: "'h list" shows "list_all2 Sb (map i x) y = list_all2 (\<lambda>x. Sb (i x)) x y"
a_minus_def: fixes R :: "('h, 'i) ring_scheme"   and x :: "'h"   and y :: "'h" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
separation_normal_local: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "S \<inter> T = {}" obtains S' :: "'a set" and T' :: "'a set" where "openin (top_of_set U) S'" and "openin (top_of_set U) T'" and "S \<subseteq> S'" and "T \<subseteq> T'" and "S' \<inter> T' = {}"
sublinear_subadd: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "sublinear t"   and "feasible t" shows "sub_add t"
divide_out_unique_nat_2: fixes p :: "nat"   and m :: "nat"   and n :: "nat"   and k :: "nat" assumes "\<not> is_unit p"   and "p \<noteq> 0"   and "\<not> p dvd m"   and "n = p ^ k * m" shows "k = multiplicity p n"
gen_bound_typ_instance: fixes S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list"   and t :: "typ" shows "gen ($ S A) ($ S t) \<le> $ S (gen A t)"
is_none_code_2: fixes x :: "'g" shows "Option.is_none (Some x) = False"
assms: shows "st \<in> reachable (pnet np (p1 \<parallel> p2)) I"
assms_2: shows "(B, C) \<in> ns_mul_ext W S"
getPos_NegPos_map_simp: fixes X :: "'b \<Rightarrow> 'a"   and src :: "'b list" shows "getPos (NegPos_map X (map Pos src)) = map X src"
enc_prop_nec_1: fixes v :: "i"   and x :: "\<nu>" shows "[\<^bold>\<diamond>(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) \<^bold>\<rightarrow> (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
scaleC_left_mono: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and a :: "complex" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "less_eq x y"   and "0 \<le> a" shows "less_eq (scaleC a x) (scaleC a y)"
length_nat_to_bv_upper_limit: fixes n :: "nat"   and k :: "nat" assumes "n \<le> 2 ^ k - 1" shows "length (nat_to_bv n) \<le> k"
simps_3: fixes a :: "poscheckvariant"   and t :: "('a, 'b) Term.term"   and t' :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = \<langle>a: (t \<cdot> \<theta>) \<doteq> t' \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t"
suminf_exist_split2: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "(\<lambda>n. \<Sum>k. f (k + n)) \<longlonglongrightarrow> (0::'b)"
simps_4: shows "subcomp sA4 = {sA41, sA42}"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"
RedStaticCall: 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 Cs :: "char list list"   and C :: "char list"   and Cs'' :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and Ds :: "char list list"   and vs :: "val list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs'' "   and "P \<turnstile> C has least M = (Ts, T, pns, body) via Cs'"   and "Ds = (Cs @\<^sub>p Cs'') @\<^sub>p Cs'"   and "length vs = length pns"   and "length Ts = length pns" shows "P,E \<turnstile> \<langle>Call (ref (a, Cs)) \<lfloor>C\<rfloor> M (map Val vs),s\<rangle> \<rightarrow> \<langle>blocks (this # pns, Class (last Ds) # Ts, Ref (a, Ds) # vs, body),s\<rangle>"
less_le_not_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
fmdom_def: shows "fmdom \<equiv> (fmlookup ---> Abs_fset) dom"
bind_option_simps_1: fixes fail :: "'c"   and f :: "'d \<Rightarrow> 'c" shows "monad.bind_option fail None f = fail"
null_def: shows "null = (THE n. \<forall>f. n \<cdot> f = n \<and> f \<cdot> n = n)"
simplex_map_eq: fixes p :: "nat"   and X :: "'c topology"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "singular_simplex p X c"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "simplex_map p f c = simplex_map p g c"
act_distribD: fixes p :: "'b"   and q :: "'b"   and x :: "'a" shows "(D p q \<and> Dla x (p \<oplus> q)) = (Dla x p \<and> Dla x q \<and> D (\<alpha> x p) (\<alpha> x q))"
dist2: shows "distinct (map fst (map_pair (\<lambda>kv. (map_component ((+) i) (fst kv), snd kv)) xs'))"
take_bit_concat_bit_eq: fixes m :: "nat"   and n :: "nat"   and k :: "int"   and l :: "int" shows "take_bit m (concat_bit n k l) = concat_bit (min m n) k (take_bit (m - n) l)"
fls_prpart_idem: fixes f :: "'a fls" shows "fls_prpart (fls_prpart_as_fls f) = fls_prpart f"
slice1_eq_of_bl: fixes n :: "nat"   and w :: "'c word" shows "slice1 n w = of_bl (takefill False n (to_bl w))"
c_sinvar_valid_imp_no_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and G :: "'b graph" assumes "configured_SecurityInvariant m"   and "c_sinvar m G" shows "c_offending_flows m G = {}"
matrix_inf_sum_def: fixes d :: "nat"   and f :: "nat \<Rightarrow> complex mat" shows "matrix_inf_sum d f = matrix_seq.lowner_lub (matrix_sum d f)"
stable_pair_on_CD_on_CH: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "CD_on ds (fst XD_XH) = CH (snd XD_XH)"
stuff_4: shows "tyenv_wellformed mds \<Gamma> \<S> P"
isotoneI: fixes L1 :: "('d, 'e) gorder_scheme"   and L2 :: "('f, 'g) gorder_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "weak_partial_order L1"   and "weak_partial_order L2"   and "\<And>x y. \<lbrakk>x \<in> carrier L1; y \<in> carrier L1; x \<sqsubseteq>\<^bsub>L1\<^esub> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq>\<^bsub>L2\<^esub> f y" shows "isotone L1 L2 f"
Enc_Tag: fixes X :: "msg"   and Y :: "msg"   and X' :: "tag" shows "Enc X Y \<noteq> Tag X'"
1: shows "C\<^sub>1.runit' \<I> \<cdot> \<iota> = (\<r>\<^sub>1[\<I>] \<otimes>\<^sub>1 \<l>[\<I>\<^sub>1]) \<cdot> (C\<^sub>1.runit' \<I> \<otimes>\<^sub>1 C\<^sub>1.runit' \<I>)"
interval_bounds_real_1: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "Sup {a..b} = b"
simp'_1: fixes x :: "Core.term" shows "subst_bv x f = f"
disc_142: fixes x12 :: "nat" shows "is_jump (Ubx.instr.ICJump x12)"
equiintegrable_halfspace_restrictions_gt: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "F equiintegrable_on cbox a b"   and "f \<in> F"   and "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm (f x)" shows "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c < x \<bullet> i then h x else (0::'b)}) equiintegrable_on cbox a b"
mult_zero_left: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and a :: "'m" assumes "class.mult_zero times zero" shows "times zero a = zero"
meet_idem: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqinter> x = x"
homologous_rel_singular_relcycle: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S c1 c2" shows "singular_relcycle p X S c1 = singular_relcycle p X S c2"
implements_actJP_jAction: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol"   and t :: "'s Trace"   and n :: "nat"   and a :: "'a" assumes "SPR.implements jp"   and "t \<in> SPR.jkbpCn n" shows "set (SPR.actJP jp t a) = set (jAction (SPR.MCn n) t a)"
induct: fixes P :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "nat" assumes "\<And>x n. (n \<noteq> 0 \<Longrightarrow> P x ((n - 1) div 2)) \<Longrightarrow> P x n" shows "P a0 a1"
stable_decision: fixes tr :: "v_state list"   and i :: "nat"   and s :: "v_state"   and t :: "v_state"   and j :: "nat"   and p :: "process"   and v :: "val" assumes "tr \<in> beh v_TS"   and "i < length tr"   and "s = tr ! i"   and "t = tr ! (i - j)"   and "decisions s p = Some v" shows "decisions t p = Some v"
vec_hom_zero_iff: fixes x :: "int vec"   and n :: "nat" shows "(of_int_hom.vec_hom x = 0\<^sub>v n) = (x = 0\<^sub>v n)"
comparator_of_def: fixes x :: "'a"   and y :: "'a" shows "comparator_of x y = (if x < y then Lt else if x = y then Eq else Gt)"
Max_triangle_triangle_le: fixes z :: "nat" shows "triangle (Max_triangle z) \<le> z"
Lyndon_suf_le: fixes w :: "'a list"   and s :: "'a list" assumes "Lyndon w"   and "s \<le>ns w" shows "w \<le>lex s"
insert: fixes G :: "prog"   and A :: "'b triple set"   and t :: "'b triple"   and ts :: "'b triple set" assumes "G,A\<turnstile>t"   and "G,A|\<turnstile>ts" shows "G,A|\<turnstile>insert t ts"
Mismatch: fixes P :: "pi"   and V :: "residual"   and a :: "name"   and b :: "name" assumes "P \<longmapsto> V"   and "a \<noteq> b" shows "[a\<noteq>b]P \<longmapsto> V"
pwd_spy_id_sesk_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "let (s, S, A, B, C, D) = pwd_spy_id_sesk n; SK = (Some S, {A, B}, {C, D}) in pwd_spy_id_sesk_state n (S, A, B, C, D) \<subseteq> s \<and> {SesKey SK} \<in> key_sets (Pwd n) (crypts (Log -` spied s))"
simps_1: fixes all :: "String.literal"   and S :: "typ"   and T :: "typ"   and t :: "Core.term" shows "strip_all_single_var (Ct all S $ Abs T t) = (if all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then Some T else None)"
g_append_cong: fixes xs1 :: "'a list"   and xs2 :: "'a list"   and ys1 :: "'a list"   and ys2 :: "'a list" assumes "g xs1 = g xs2"   and "g ys1 = g ys2" shows "g (xs1 @ ys1) = g (xs2 @ ys2)"
f_Pi: shows "f \<in> [elts \<alpha>]\<^bsup>2\<^esup> \<rightarrow> {..<Suc (Suc 0)}"
basic_prop_3: fixes v :: "i" shows "[Contingent (q\<^sub>0\<^sup>-) in v]"
distinct_15: fixes x31 :: "'b aexp"   and x32 :: "'b aexp"   and x41 :: "'b"   and x42 :: "value list" shows "Gt x31 x32 \<noteq> In x41 x42"
abs_summable_hurwitz_zeta: fixes s :: "complex"   and a :: "real"   and b :: "nat" assumes "1 < Re s"   and "0 < a + real b" shows "(\<lambda>n. 1 / complex_of_real (real n + a) powr s) abs_summable_on {b..}"
bilinear_l_lin1: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m1 :: "'g"   and m2 :: "'i"   and r :: "'a"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m1 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "r \<in> carrier R"   and "bilinear_map f R M1 M2 N" shows "f (r \<cdot>\<^sub>s\<^bsub>M1\<^esub> m1, m2) \<plusminus>\<^bsub>N\<^esub> -\<^sub>a\<^bsub>N\<^esub> r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (m1, m2) = \<zero>\<^bsub>N\<^esub>"
wf: shows "wf {(c2, c1). \<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma>\<turnstile> c1 \<rightarrow> c2}"
geom_rand_walk_pseudo_proj_True: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and n :: "nat" assumes "prob_grw p M geom_proc u d init" shows "geom_proc n = geom_proc n \<circ> pseudo_proj_True n"
runit_eqI: fixes f :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : a \<otimes>\<^sub>C \<I>\<^sub>C \<rightarrow>\<^sub>C a\<guillemotright>"   and "f \<otimes>\<^sub>C \<I>\<^sub>C = (a \<otimes>\<^sub>C \<iota>\<^sub>C) \<cdot>\<^sub>C \<a>\<^sub>C[a, \<I>\<^sub>C, \<I>\<^sub>C]" shows "f = \<r>\<^sub>C[a]"
derangementsI: fixes p :: "nat \<Rightarrow> nat"   and S :: "nat set" assumes "p permutes S"   and "\<And>x. x \<in> S \<Longrightarrow> p x \<noteq> x" shows "p \<in> derangements S"
lprodr_def: shows "lprodr = (\<lambda>(a, b, c). ((a, b), c))"
circ_sup_4: fixes circ :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.left_zero_kleene_conway_semiring circ one times sup less_eq less bot star" shows "circ (sup x y) = times (circ (times (star x) y)) (circ x)"
sorted1: fixes x :: "'d Word.word" shows "signed.sorted [x] = True"
measurable_marg_dens': fixes x :: "nat" assumes "x \<in> V" shows "(\<lambda>(x, y). marg_dens \<Y> x x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure (\<Gamma> x))"
UN_iff: fixes b :: "'f"   and B :: "'g \<Rightarrow> 'f set"   and A :: "'g set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
eq: fixes A :: "'a set" shows "snd ` Set.filter (\<lambda>(x, y). R x y) (A \<times> set_pmf q) = {y. \<exists>x\<in>A. R x y} \<inter> set_pmf q"
map_spmf_cong: fixes p :: "'d spmf"   and q :: "'d spmf"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "map_spmf f p = map_spmf g q"
Red1FAssNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and F :: "String.literal"   and D :: "String.literal"   and v :: "'addr val"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<bullet>F{D} := Val v,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
assms_1: shows "matchs xs ys = Some env\<^sub>1"
truth: shows "- (2 ^ (size (SCAST(32 \<rightarrow> 64) w2) - 1)) \<le> sint (SCAST(32 \<rightarrow> 64) w2) + sint 0x7FFFFFFF \<and> sint (SCAST(32 \<rightarrow> 64) w2) + sint 0x7FFFFFFF \<le> 2 ^ (size (SCAST(32 \<rightarrow> 64) w2) - 1) - 1"
simps_1: fixes p :: "char list"   and ts :: "fterm list"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "Pos p ts \<cdot>\<^sub>l \<sigma> = Pos p (ts \<cdot>\<^sub>t\<^sub>s \<sigma>)"