hcju/isabelleps huggingface dataset:
rref_condition1_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" assumes "reduced_row_echelon_form A"   and "is_zero_row i A" shows "\<forall>j>i. is_zero_row j A"
exewf_sort_def: fixes sub :: "(String.literal \<times> String.literal) list"   and S :: "String.literal set" shows "exewf_sort sub S \<equiv> exenormalized_sort sub S \<and> exesort_ex sub S"
cosafe_rpd: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Past Lax r"   and "s \<in> rpd test i r" shows "local.safe_regex Past Lax s"
nth_default_strip_while_dflt: fixes dflt :: "'b"   and xs :: "'b list" shows "nth_default dflt (strip_while ((=) dflt) xs) = nth_default dflt xs"
small_refl_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f \<in> l F f) = (\<forall>\<^sub>F x in F. f x = (0::'b))"
finite_enumerate_Ex: fixes S :: "'a set"   and s :: "'a" assumes "finite S"   and "s \<in> S" shows "\<exists>n<card S. wellorder_class.enumerate S n = s"
appliedBacktrack_def: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list" shows "appliedBacktrack stateA stateB F0 \<equiv> formulaFalse F0 (elements (getM stateA)) \<and> decisions (getM stateA) \<noteq> [] \<and> getM stateB = prefixBeforeLastDecision (getM stateA) @ [(opposite (lastDecision (getM stateA)), False)]"
acute__not_obtuse: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C" shows "\<not> Tarski_neutral_dimensionless.Obtuse Bet Cong A B C"
rcliff_greek_less2: fixes r :: "('a \<times> 'a) set"   and as :: "(accent \<times> 'a) list"   and a :: "'a"   and b' :: "(accent \<times> 'a) list"   and b :: "'a"   and cs :: "(accent \<times> 'a) list" assumes "trans r"   and "snd ` set as \<subseteq> under r a"   and "b' \<in> {[(Grave, b)], []}"   and "snd ` set cs \<subseteq> under r a \<union> under r b" shows "(as @ b' @ cs, [(Macron, a), (Grave, b)]) \<in> greek_less r"
bilinear_rzero: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a" assumes "bilinear h" shows "h x (0::'b) = (0::'c)"
the_\<nu>_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" assumes "arrow_of_spans_of_maps V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s\<^sub>0 s\<^sub>1 w" shows "src (arrow_of_spans_of_maps.the_\<nu> V H r\<^sub>1 s\<^sub>1 w) = src r\<^sub>1"
proj_image: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Equiv_Relations.proj r ` A = A // r"
take_nemp: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>"   and "n \<noteq> 0" shows "take n w \<noteq> \<epsilon>"
smult_r_distr: fixes x :: "'a vec"   and y :: "'a vec"   and a :: "'a" assumes "True"   and "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "a \<cdot>\<^sub>v (x + y) = a \<cdot>\<^sub>v x + a \<cdot>\<^sub>v y"
less_diff_imp_less: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i < j - m" shows "i < j"
IH2: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE2}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"   and "osafe ODE2" shows "ODE_sem (local.adjoint I \<sigma> \<nu>) ODE2 = ODE_sem (local.adjoint I \<sigma> \<omega>) ODE2"
max_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
sum_moebius_mu_sublinear_imp_PNT: assumes "M \<in> o(\<lambda>x. x)" shows "\<psi> \<sim>[at_top] (\<lambda>x. x)"
simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c"   and t :: "'c"   and ts :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f (t # ts) = term.list_comb app (app f t) ts"
has_locks_acquire_locks_conv: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and n :: "nat" assumes "may_lock l t" shows "has_locks (acquire_locks l t n) t = has_locks l t + n"
comp_left_mono: fixes S :: "'e \<Rightarrow> 'f"   and S' :: "'e \<Rightarrow> 'f"   and T :: "'g \<Rightarrow> 'e" assumes "S \<sqsubseteq> S'" shows "S \<circ> T \<sqsubseteq> S' \<circ> T"
assms_4: assumes "x \<le> 0"   and "y \<le> 0" shows "P (lcm (- x) (- y))"
in_homI: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.arr f"   and "C.VV.dom f = a"   and "C.VV.cod f = b" shows "C.VV.in_hom f a b"
is_subspace_of_vec_list_def: fixes n :: "nat"   and vs :: "'b vec list"   and ws :: "'b vec list" shows "is_subspace_of_vec_list n vs ws = (let ws' = gram_schmidt0 n ws in \<forall>v\<in>set vs. adjuster n v ws' = - v)"
add_less_same_cancel2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus a b) b = less a zero"
prj_chine_1: shows "THfgh.p\<^sub>0 \<star> TTfgh_THfgh.chine \<cong> h.chine \<star> TTfgh.p\<^sub>0"
less_dual_def_var: fixes y :: "'a"   and x :: "'a" shows "(\<partial> y < \<partial> x) = (x < y)"
SumPres': fixes P :: "pi"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<equiv>\<^sub>e P'"   and "Q \<equiv>\<^sub>e Q'" shows "P \<oplus> Q \<equiv>\<^sub>e P' \<oplus> Q'"
is_empty_stack_def: fixes s :: "('v, 'es) state_scheme" shows "is_empty_stack s \<equiv> stack s = []"
bij_extends: fixes p :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "bij p"   and "p x = x" shows "bij (p(x := y, inv p y := x))"
inverse_unique: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "inverse_arrows f g" shows "local.inv f = g"
neg_less_minus_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "c < 0" shows "(a < - (b /\<^sub>R c)) = (- b < c *\<^sub>R a)"
arr_iff_in_hom: fixes f :: "'a" shows "hom.arr f = hom.in_hom f (hom.dom f) (hom.cod f)"
dfa_Power: fixes M :: "'b nfa" assumes "nfa M" shows "dfa (nfa.Power_dfa M)"
FVTsub1: assumes "vid1 \<in> ODE_vars I (OVar vid1)" shows "FVT ($f fid2 (\<lambda>i. if i = vid1 then Var vid1 else Const 0)) \<subseteq> semBV I (OVar vid1)"
simps_1: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat" shows "steps c p 0 = c"
round_sort_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat"   and r :: "nat" assumes "index_less index key"   and "index_mono index key"   and "bn_inv p q t"   and "add_inv n t"   and "sort_inv key t" shows "sort_inv key (round index key p q r t)"
cycle_altdef: fixes p :: "'b list" shows "cycle p = (closed_w p \<and> (\<exists>u. distinct (tl (awalk_verts u p))))"
pnorm_def: shows "pnorm = flatten PLUS \<circ> pset"
integral_suminf: fixes M :: "'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. integrable M (f i)"   and "AE x in M. summable (\<lambda>i. norm (f i x))"   and "summable (\<lambda>i. LINT x|M. norm (f i x))" shows "LINT x|M. (\<Sum>i. f i x) = (\<Sum>i. integral\<^sup>L M (f i))"
strict_not_refl_weak: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier" shows "((y, x) \<notin> relation) = (x \<succeq>[relation] y \<and> (y, x) \<notin> relation)"
con_rews_4: fixes b :: "'c" shows "(Right\<cdot>b = \<bottom>) = (b = \<bottom>)"
S2P_def: fixes A :: "'b set" shows "S2P A \<equiv> \<lambda>x. x \<in> A"
additive_principal_oLimit: fixes f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. additive_principal (f n)" shows "additive_principal (oLimit f)"
Fun_pr0: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "Fun (pr0 a b) = (\<lambda>x\<^bold>\<in>local.set (local.prod a b). UP (hsnd (DOWN x)))"
strict_iff_order: fixes b :: "'g"   and a :: "'g" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
OneState_HPT_Target_ChiRel: 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 U :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) seqauto"   and S :: "'d" assumes "TS \<in> HPT ST"   and "(U, T) \<in> ChiRel (HA ST)"   and "U \<in> Target TS"   and "A \<in> SAs (HA ST)"   and "T \<in> States A"   and "S \<in> States A" shows "S \<notin> Target TS"
intyness_numeral: fixes num :: "num" assumes "num \<equiv> num" shows "numeral num \<equiv> real (numeral num)"
simps: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'd" shows "integrable M f = (\<exists>x. has_bochner_integral M f x)"
conjug: shows "\<rho> u \<sim> \<rho> v"
isodefl_imp_deflation: fixes d :: "'a \<rightarrow> 'a"   and t :: "udom defl" assumes "isodefl d t" shows "deflation d"
module: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme" assumes "LinearCombinations.submodule R N M" shows "module R M"
indspaceI': fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set"   and G :: "'g set" assumes "supp f \<subseteq> FH"   and "range f \<subseteq> V"   and "\<forall>x\<in>FH. \<forall>y\<in>FH. f (x + y) = f x + f y"   and "\<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x" shows "f \<in> indspace G FH V"
d_conc_s_prod_ax: fixes x :: "'a"   and y :: "'a" shows "d x \<parallel> d y = d x \<cdot> d y"
Perp_in_perm: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt A B C D  \<and> X PerpAt B A C D  \<and> X PerpAt A B D C  \<and> X PerpAt B A D C  \<and> X PerpAt C D A B  \<and> X PerpAt C D B A  \<and> X PerpAt D C A B  \<and> X PerpAt D C B A "
neg_def: fixes \<phi> :: "'c fm" shows "neg \<phi> = (if \<phi> = TrueF then FalseF else if \<phi> = FalseF then TrueF else Neg \<phi>)"
alg_mult_gt_zero_iff_is_root: fixes p :: "'a list list"   and x :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))" shows "(0 < UP.alg_mult p x) = UP.is_root p x"
antisym_conv1: fixes x :: "'c list"   and y :: "'c list" assumes "\<not> strict_prefix x y" shows "prefix x y = (x = y)"
group_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c" shows "Groups.group f z inverse \<equiv> semigroup f \<and> Groups.group_axioms f z inverse"
induct: fixes P1 :: "ty \<Rightarrow> ty \<Rightarrow> bool"   and P2 :: "funty \<Rightarrow> funty \<Rightarrow> bool"   and x1 :: "ty"   and x2 :: "ty"   and x3 :: "funty"   and x4 :: "funty" assumes "\<And>A. P1 A A"   and "\<And>f1 f2. \<lbrakk>f1 <:: f2; P2 f1 f2\<rbrakk> \<Longrightarrow> P1 (TFun f1) (TFun f2)"   and "\<And>T1 T1' T2 T2'. \<lbrakk>T1 <: T1'; P1 T1 T1'; T1' <: T1; P1 T1' T1; T2 <: T2'; P1 T2 T2'; T2' <: T2; P1 T2' T2\<rbrakk> \<Longrightarrow> P2 (T1 \<rightarrow> T2) (T1' \<rightarrow> T2')"   and "\<And>T1 T2. P2 (T1 \<sqinter> T2) T1"   and "\<And>T1 T2. P2 (T1 \<sqinter> T2) T2"   and "\<And>T3 T1 T2. \<lbrakk>T3 <:: T1; P2 T3 T1; T3 <:: T2; P2 T3 T2\<rbrakk> \<Longrightarrow> P2 T3 (T1 \<sqinter> T2)"   and "\<And>T1 T2. P2 (T1 \<rightarrow> T2) \<top>"   and "P2 \<top> \<top>"   and "\<And>T. P2 T T"   and "\<And>T1 T2 T3. \<lbrakk>T1 <:: T2; P2 T1 T2; T2 <:: T3; P2 T2 T3\<rbrakk> \<Longrightarrow> P2 T1 T3" shows "(x1 <: x2 \<longrightarrow> P1 x1 x2) \<and> (x3 <:: x4 \<longrightarrow> P2 x3 x4)"
lconjby_add: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "lconjby (x + y) z = lconjby x (lconjby y z)"
vec_index_le_linf_norm: fixes i :: "nat"   and v :: "'b vec" assumes "i < dim_vec v" shows "\<bar>v $ i\<bar> \<le> \<parallel>v\<parallel>\<^sub>\<infinity>"
ccProd_insert1: fixes x :: "Vars.var"   and S' :: "Vars.var set"   and S :: "Vars.var set" shows "insert x S' G\<times> S = {x} G\<times> S \<squnion> S' G\<times> S"
fst_conv: fixes x1 :: "'a"   and x2 :: "'b" shows "cc (x1, x2) = x1"
N_def: shows "N = {s \<in> S. n s = 0} \<union> S2"
cscc_invarI: fixes v0 :: "'v"   and D0 :: "'v set"   and PDPE :: "'v set list \<times> 'v set \<times> ('v \<times> 'v) set"   and l :: "'v set list" assumes "invar v0 D0 PDPE"   and "invar v0 D0 PDPE \<Longrightarrow> cscc_invar_part (l, PDPE)" shows "cscc_invar v0 D0 (l, PDPE)"
inject: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b"   and y21 :: "'a"   and y22 :: "'a"   and y23 :: "'b" shows "(J_C.MkArr x21 x22 x23 = J_C.MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
permutation_inverse_works_1: fixes p :: "'c \<Rightarrow> 'c" assumes "permutation p" shows "inv p \<circ> p = id"
typedef_bi_unique: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "bi_unique T"
prems_2: shows "i \<notin> list_nominals (p # ps)"
sel_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (ShuffleLR r s) = (\<oo> r \<and> \<oo> s)"
UP_car_memI: fixes k :: "nat"   and p :: "nat \<Rightarrow> 'a" assumes "\<And>n. k < n \<Longrightarrow> p n = \<zero>"   and "\<And>n. p n \<in> carrier R" shows "p \<in> carrier P"
floor_of_nat: fixes n :: "nat" shows "\<lfloor>of_nat n\<rfloor> = int n"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VxV.comp g f = (if B.seq (fst g) (fst f) \<and> B.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>B fst f, snd g \<cdot>\<^sub>B snd f) else (B.null, B.null))"
locally_finite_on_open_coverI: fixes I :: "'c set"   and U :: "'c \<Rightarrow> 'd set"   and X :: "'d set" assumes "\<And>j. j \<in> I \<Longrightarrow> finite {i \<in> I. U i \<inter> U j \<noteq> {}}"   and "X \<subseteq> \<Union> (U ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> open (U i)" shows "locally_finite_on X I U"
interp_msg_def: fixes msg :: "'a \<times> ('a, 'b) operation"   and state :: "'b \<Rightarrow> 'a set" shows "interp_msg msg state \<equiv> \<langle>snd msg\<rangle> state"
Sup_inf: fixes B :: "'a set"   and a :: "'a" shows "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>\<eta> f : D.map a \<rightarrow>\<^sub>D GF.map b\<guillemotright>"
Or_prop_entailment: fixes S :: "'c ltl set"   and xs :: "'c ltl list" shows "S \<Turnstile>\<^sub>P Or xs = (\<exists>x\<in>set xs. S \<Turnstile>\<^sub>P x)"
b: fixes n :: "nat" shows "(\<lambda>k. \<Sum>i<k. \<integral>\<^sup>+x\<in>D (n + i). f ((T ^^ (i + 1)) x)\<partial>M) \<longlonglongrightarrow> (\<Sum>i. d (i + 1) (n + i))"
assms_2: shows "cfg' \<in> set_pmf (K_cfg (repcs (l, u) cfg))"
seqI: fixes f :: "'a"   and g :: "'a" assumes "C.arr f"   and "C.arr g"   and "C.dom g = C.cod f" shows "C.seq g f"
no_overlap_not_unefined: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list"   and p :: "'e" assumes "check_no_overlap \<gamma> ft" shows "OF_same_priority_match2 \<gamma> ft p \<noteq> Undefined"
strict_mono_less_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x \<le> f y) = less_eq x y"
IH: fixes s :: "'state" assumes "exit s (hd vl)"   and "validFrom s tr"   and "V tr = vl"   and "never T tr" shows "False"
returns_result_heap_def: fixes h :: "'a"   and p :: "('a, 'b, 'c) prog"   and r :: "'c"   and h' :: "'a" shows "h \<turnstile> p \<rightarrow>\<^sub>r r \<rightarrow>\<^sub>h h' = (h \<turnstile> p \<rightarrow>\<^sub>r r \<and> h \<turnstile> p \<rightarrow>\<^sub>h h')"
assms_1: shows "Unifier \<sigma> (s \<cdot> \<I>) (t \<cdot> \<I>)"
less_top_def: fixes x :: "'b top"   and y :: "'b top" shows "(x < y) = (case x of top.Value x \<Rightarrow> case y of top.Value y \<Rightarrow> x < y | top.Top \<Rightarrow> True | top.Top \<Rightarrow> False)"
greaterThanLessThan_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and k :: "'b" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanLessThan less k l = {}"
domain_unique: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" assumes "a \<in> VV.domains f"   and "a' \<in> VV.domains f" shows "a = a'"
es: shows "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')"
assms: shows "(\<lambda>i. measure_pmf.prob N {i..}) abs_summable_on {1..}"
Neg_Melem_subst_atm_subst_cls: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and A :: "'b"   and C :: "'b literal multiset"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "Neg A \<in># C" shows "Neg (subst_atm A \<sigma>) \<in># substitution_ops.subst_cls subst_atm C \<sigma>"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "C.ide b"   and "C.seq b f" shows "b \<cdot>\<^sub>C f = f"
local_flow_ball: fixes g :: "real" shows "local_flow (fball g) UNIV UNIV (\<lambda>t s. \<chi>i. if i = 1 then g * t\<^sup>2 / 2 + s $ 2 * t + s $ 1 else g * t + s $ 2)"
conc_trans_additional_3: fixes A :: "'b nres"   and R :: "('b \<times> 'c) set"   and B :: "'c nres"   and C :: "'c nres" assumes "A \<le> \<Down> R B"   and "B \<le> \<Down> Id C" shows "A \<le> \<Down> R C"
the_inv_into_f_f: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
erase2jumpover1: fixes q :: "nat"   and list :: "nat list" assumes "q < length list"   and "\<forall>rn. <drop q list> \<noteq> Oc # Oc \<up> list ! q @ Bk # <drop (Suc q) list> @ Bk \<up> rn" shows "<drop q list> = Oc # Oc \<up> list ! q"
inject_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Broadcast x1 = Broadcast y1) = (x1 = y1)"
goal_assms: shows "\<psi> \<in> new n \<and> (\<exists>\<nu> \<mu>. \<psi> = \<nu> or\<^sub>r \<mu> \<or> \<psi> = \<nu> U\<^sub>r \<mu> \<or> \<psi> = \<nu> R\<^sub>r \<mu>)"
case_eq_if: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "(case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = \<bottom> then f1 else f2 (the option))"
mult_nonneg_nonpos2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times b a) zero"
eq_iff: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(x = y) = (x \<subseteq># y \<and> y \<subseteq># x)"
AB_2: shows "normalize (\<Prod>\<^sub># A) = normalize x"
elims_3: fixes x :: "('d \<times> 'e \<times> 'd) set \<times> 'd \<times> (('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and xa :: "nat \<Rightarrow> 'e" assumes "\<not> accept\<^sub>G\<^sub>R_LTS x xa" obtains \<delta> :: "('d \<times> 'e \<times> 'd) set" and q\<^sub>0 :: "'d" and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set" and w :: "nat \<Rightarrow> 'e" where "x = (\<delta>, q\<^sub>0, \<alpha>)" and "xa = w" and "\<not> (\<exists>(Fin, Inf)\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 (Fin, Inf) w)"
vsv_vconverse: shows "vsv ((r\<inverse>\<^sub>\<circ>)\<inverse>\<^sub>\<circ>)"
run_altc_sample_option: fixes altc_sample :: "'b \<Rightarrow> ('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'c \<Rightarrow> ('a, 'm) optionT" shows "run_option (local.altc_sample_option altc_sample p f) = altc_sample p (\<lambda>x. run_option (f x))"
is_unifiers_comp: fixes \<sigma> :: "'s"   and As :: "'a list"   and Bs :: "'a multiset list"   and \<eta> :: "'s" shows "is_unifiers \<sigma> (set_mset ` set (Map2.map2 add_mset As Bs) \<cdot>ass \<eta>) = is_unifiers (\<eta> \<odot> \<sigma>) (set_mset ` set (Map2.map2 add_mset As Bs))"
per23_preserves_bet_inv: fixes A :: "'p"   and B' :: "'p"   and C' :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B' C'"   and "A \<noteq> B'"   and "Col A B C"   and "Per A B' B"   and "Per A C' C" shows "Bet A B C"
right_neutral: fixes a :: "'k" shows "a * (1::'k) = a"
A: fixes M :: "real" shows "\<forall>\<^sub>F n in sequentially. ereal M \<le> extended_Gromov_product_at basepoint (to_Gromov_completion (v n)) x"
not_empty_eq_Iic_eq_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c" assumes "class.preorder less_eq less" shows "{} \<noteq> ord.atMost less_eq h"
spmf_bind: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf"   and y :: "'a" shows "spmf (bind_spmf p f) y = LINT x|measure_spmf p. spmf (f x) y"
less_imp_le: fixes x :: "'g"   and y :: "'g" assumes "x < y" shows "x \<le> y"
det_multrow: fixes k :: "nat"   and n :: "nat"   and A :: "'b mat"   and a :: "'b" assumes "k < n"   and "A \<in> carrier_mat n n" shows "det (multrow k a A) = a * det A"
sel_1: fixes x11 :: "char list"   and x12 :: "'a list" shows "get_pred (Pos x11 x12) = x11"
conseq: fixes P :: "'b assn"   and G :: "'b triple set"   and c :: "com"   and Q :: "'b assn" assumes "\<forall>Z s. P Z s \<longrightarrow> (\<exists>P' Q'. G|-{P'}. c .{Q'} \<and> (\<forall>s'. (\<forall>Z'. P' Z' s \<longrightarrow> Q' Z' s') \<longrightarrow> Q Z s'))" shows "G|-{P}. c .{Q}"
base: fixes C :: "'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "([], C) \<in> R" shows "(C, 0) \<in> derivable R"
sort_by_mergesort_rec: fixes xs :: "'a list" shows "Sorting_Algorithms.sort cmp xs = local.merge (Sorting_Algorithms.sort cmp (take (length xs div 2) xs)) (Sorting_Algorithms.sort cmp (drop (length xs div 2) xs))"
assms_2: assumes "a" shows "Case_Labeling.CTXT (Suc inp) (ct' inp ''then'') outp' b"
subformulas\<^sub>\<nu>_finite: fixes \<phi> :: "'a ltln" shows "finite (subformulas\<^sub>\<nu> \<phi>)"
abs_eq: shows "0 = Bcontfun (\<lambda>_. 0::'d)"
nc: shows "\<not> B $$ (0, 0) * e \<le> c"