hcju/isabelleps huggingface dataset:
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> y \<in> carrier V"
has_derivative_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
term_subst_eq_conv: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "(t \<cdot> \<sigma> = t \<cdot> \<tau>) = (\<forall>x\<in>vars_term t. \<sigma> x = \<tau> x)"
simps: fixes s :: "('a, 'b option) insert_message list"   and i :: "'a extended" shows "idx s i = fromSingleton (filter (\<lambda>j. ext_ids s ! j = i) [0..<length (ext_ids s)])"
\<mu>'_eq: fixes A :: "'key set" assumes "A \<subseteq> keys" shows "K.prob A = sum K A"
derivation_shift_right_left_subtract: fixes left :: "nat"   and right :: "nat"   and L :: "(nat \<times> symbol \<times> symbol list) list" assumes "left \<le> right" shows "derivation_shift (derivation_shift L 0 right) left 0 = derivation_shift L 0 (right - left)"
usubstappp_loop: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" shows "usubstappp \<sigma> U \<alpha>** = (fst (usubstappp \<sigma> U \<alpha>), Loopo (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>)))"
ReachCfg2Cfg3: shows "reachable cfg2 cfg3"
compute_None_Some_eq: fixes x :: "'a" shows "(None = Some x) = False"
compose_id_inv_into: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" assumes "f ` A = B" shows "compose B f (restrict (inv_into A f) B) = (\<lambda>x\<in>B. x)"
sets_tree_sigma_count_space: fixes B :: "'b set" assumes "countable B" shows "sets (tree_sigma (count_space B)) = Pow (trees B)"
iWeakUntil_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iWeakUntil I P Q \<equiv> (\<box> t I. P t) \<or> (\<diamond> t I. Q t \<and> (\<box> t' I \<down>< t. P t'))"
Compl_iff: fixes c :: "'a"   and A :: "'a set" shows "(c \<in> - A) = (c \<notin> A)"
comp_runit_term_dup: fixes a :: "'a" assumes "ide a" shows "runit a \<cdot> local.prod a \<t>[a] \<cdot> dup a = a"
enn2real_eq_0_iff: fixes x :: "ennreal" shows "(enn2real x = 0) = (x = 0 \<or> x = \<top>)"
assms_1: shows "Seed\<^bsub>p\<^esub> [next_plane\<^bsub>p\<^esub>]\<rightarrow>* g"
natural_transformation_axioms: shows "natural_transformation C.CC.comp C.CC.comp C.CC.map C.CC.map C.CC.map"
invfun_r1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and b :: "'d" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "b \<in> B" shows "f ((f\<inverse>\<^bsub>B,A\<^esub>) b) = b"
discrete_imp_closed: fixes e :: "real"   and S :: "'b set" assumes "0 < e"   and "\<forall>x\<in>S. \<forall>y\<in>S. dist y x < e \<longrightarrow> y = x" shows "closed S"
reflp_leq_lit: fixes A :: "('a, 'b) Term.term literal set" shows "reflp_on leq_lit A"
that: fixes s :: "real" assumes "is_Arg z s"   and "s \<in> R"   and "\<And>t. \<lbrakk>is_Arg z t; t \<in> R\<rbrakk> \<Longrightarrow> s = t" shows "thesis"
tree_height_estimate_upper: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tree_invar t" shows "size (SkewBinomialHeapStruc.tree_to_multiset t) \<le> 2 ^ Suc (SkewBinomialHeapStruc.height_tree t) - 1"
swap: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and B :: "'e set"   and A :: "'d set" shows "(\<Sum>i\<in>A. sum (g i) B) = (\<Sum>j\<in>B. \<Sum>i\<in>A. g i j)"
P: shows "signed_take_bit (size a - 1) (sint a * sint b) \<in> range (signed_take_bit (size a - 1))"
prefs_undefined: fixes i :: "'agent"   and x :: "'alt"   and y :: "'alt" assumes "i \<notin> agents" shows "\<not> R i x y"
fresh_aux: fixes c :: "name"   and a :: "name"   and x :: "'b"   and b :: "name" assumes "c \<noteq> a"   and "a \<sharp> x"   and "c \<sharp> x" shows "c \<sharp> [(a, b)] \<bullet> x"
n_1: shows "enat n < llength P \<and> P $ n \<notin> A - W"
Interp_imp_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and D' :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D"   and "D \<le> D'"   and "ground_resolution_with_selection.Interp S N D \<Turnstile> C" shows "ground_resolution_with_selection.Interp S N D' \<Turnstile> C"
assms_4: shows "D = 1\<^sub>m (n - m)"
comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c"   and i :: "'c \<Rightarrow> 'a" shows "map_endofun f g \<circ> map_endofun h i = map_endofun (f \<circ> h) (i \<circ> g)"
rotate_append: fixes l :: "'a list"   and q :: "'a list" shows "rotate |l| (l @ q) = q @ l"
mod_prod_eq: fixes f :: "'c \<Rightarrow> 'a"   and a :: "'a"   and A :: "'c set" shows "(\<Prod>i\<in>A. f i mod a) mod a = prod f A mod a"
closedin_topspace: fixes U :: "'d topology" shows "closedin U (topspace U)"
l2_inv2_trans: shows "{l2_inv2} TS.trans l2 {> l2_inv2}"
prems_2: shows "set (r # rs) \<subseteq> R"
scalar_prod_vCons: fixes a :: "'a"   and v :: "'a vec"   and b :: "'a"   and w :: "'a vec" shows "vCons a v \<bullet> vCons b w = a * b + v \<bullet> w"
StaticUpCast: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Ds),s\<^sub>1\<rangle>"
assms: shows "\<S>\<bullet>\<C> \<turnstile> es @ [e] : t1s _> t3s"
case_2: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and x2 :: "'f" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
ord_canc_left: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u + s \<preceq> u + t" shows "s \<preceq> t"
closed_segment_subset_domainI: fixes t0 :: "real"   and t :: "real"   and s :: "real" assumes "t0 \<in> T"   and "t \<in> T"   and "s \<in> {t0--t}" shows "s \<in> T"
cint_fxy: fixes y :: "complex" assumes "y \<notin> path_image \<gamma>" shows "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>"
mult_smult_right: fixes p :: "'c poly"   and a :: "'c"   and q :: "'c poly" shows "p * Polynomial.smult a q = Polynomial.smult a (p * q)"
divides_prod_r: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a divides\<^bsub>poly_ring R\<^esub> b"   and "a \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)" shows "a divides\<^bsub>poly_ring R\<^esub> b \<otimes>\<^bsub>poly_ring R\<^esub> c"
1: shows "lookup_pp (rep_nat_pp t) x < lookup_pp (rep_nat_pp s) x"
orthogonal_transformation_bij: fixes f :: "'a \<Rightarrow> 'a" assumes "orthogonal_transformation f" shows "bij f"
typeof_simps_5: fixes c :: "CTypeId"   and a :: "nat" shows "\<tau> objV c a = CClassT c"
stream_dec_not_Inl: fixes n :: "nat"   and FO :: "nat set"   and s :: "('a \<times> bool list) stream"   and i :: "nat"   and p :: "nat" assumes "stream_dec n FO s ! i = Inl p"   and "i \<notin> FO"   and "i < n" shows "False"
annot_correct: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.annot t = sum_list (map snd (FingerTreeStruc.toList t))"
decr_maxindex: fixes p :: "'b poly" assumes "polybound0 p" shows "maxindex (decrpoly p) = maxindex p - 1"
cases: fixes x :: "tm" obtains z :: "name" and r :: "tm" and z' :: "name" and xa :: "name" and y :: "name" and x' :: "name" and y' :: "name" where "atom z \<sharp> (r, z', xa, y, x', y')" and "atom z' \<sharp> (r, xa, y, x', y')" and "atom xa \<sharp> (r, y, x', y')" and "atom y \<sharp> (r, x', y')" and "atom x' \<sharp> (r, y')" and "atom y' \<sharp> r" and "x = r"
succ_ex_2: fixes R :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>" shows "succ \<R> R \<in> Succ \<R> R"
assms_12: shows "InvariantEquivalentZL (getF state) (getM state) F0"
lambda\<Pi>\<^sub>1_aux: fixes F :: "\<Pi>\<^sub>1" shows "make\<Pi>\<^sub>1 (\<lambda>u s w. \<exists>x. \<nu>\<upsilon> x = u \<and> eval\<Pi>\<^sub>1 F (\<nu>\<upsilon> x) s w) = F"
ASSUME_rule: fixes \<Phi> :: "bool"   and \<Psi> :: "unit \<Rightarrow> bool" assumes "\<Phi> \<Longrightarrow> \<Psi> ()" shows "ASSUME \<Phi> \<le> SPEC \<Psi>"
inv_Q: shows "invertible_mat (Matrix.mat 2 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then s else if i = 0 \<and> j = 1 then - b1 else if i = 1 \<and> j = 0 then t else a1))"
carrier_matI: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "dim_row A = nr"   and "dim_col A = nc" shows "A \<in> carrier_mat nr nc"
mem_set_imp_le_max_list: fixes xs :: "'b list"   and a :: "'b" assumes "\<And>a b. \<lbrakk>a \<in> set xs; b \<in> set xs\<rbrakk> \<Longrightarrow> a \<le> b \<or> b \<le> a"   and "a \<in> set xs" shows "a \<le> Missing_Lemmas.max_list xs"
R1: fixes p :: "'a"   and q :: "'a" shows "H p (R p q) q"
s_prod_annil: fixes x :: "('a \<times> 'a set) set" shows "{} \<cdot> x = {}"
filterlim_subseq: fixes f :: "nat \<Rightarrow> nat" assumes "strict_mono f" shows "filterlim f sequentially sequentially"
WHATWHERE_Secure_Assign: fixes x :: "'id"   and \<iota> :: "nat"   and e :: "'exp" assumes "e \<equiv>\<^bsub>DA x,htchLoc \<iota>\<^esub> e"   and "\<forall>m m' d \<iota>'. m \<sim>\<^bsub>d,htchLoc \<iota>'\<^esub> m' \<and> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m) =\<^bsub>d\<^esub> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m') \<longrightarrow> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m) \<sim>\<^bsub>d,htchLoc \<iota>'\<^esub> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m')" shows "WHATWHERE_Secure [x:=\<^bsub>\<iota>\<^esub> e]"
goal_def: fixes pad :: "nat \<Rightarrow> 'a vec \<Rightarrow> 'a vec"   and unpad :: "nat \<Rightarrow> 'a vec \<Rightarrow> 'a vec"   and W :: "'a vec set" shows "goal pad unpad W \<equiv> pad m (N.lincomb a W) = NM.lincomb (a \<circ> unpad m) (pad m ` W)"
poly_roots_dvd: fixes p :: "'a poly"   and n :: "nat"   and q :: "'a poly" assumes "p \<noteq> 0"   and "degree p = n"   and "n \<le> card {x. poly p x = (0::'a)}"   and "{x. poly p x = (0::'a)} \<subseteq> {x. poly q x = (0::'a)}" shows "p dvd q"
even_plus_one_iff: shows "even (x + 1) = odd x"
filter_fset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a fset" shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
m0: shows "(if 0 = 0 then mat_extension dims vars2 M0 else if 0 = 1 then mat_extension dims vars2 M1 else undefined) = exM0"
span_subspace: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and B :: "'e set" assumes "module scale"   and "A \<subseteq> B"   and "B \<subseteq> module.span scale A"   and "module.subspace scale B" shows "module.span scale A = B"
exec: shows "\<Gamma>\<turnstile> \<langle>c,Abrupt t\<rangle> \<Rightarrow> Abrupt t"
assms_1: shows "((\<lambda>x. norm (f x / g x)) \<longlongrightarrow> c) F"
dist_trans: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and d1 :: "nat"   and w :: "nat"   and d2 :: "nat"   and v :: "nat" assumes "Graph.dist c u d1 w"   and "Graph.dist c w d2 v" shows "Graph.dist c u (d1 + d2) v"
powsum_mpoly_nz: fixes X :: "nat set"   and k :: "nat" assumes "finite X"   and "X \<noteq> {}"   and "0 < k" shows "powsum_mpoly X k \<noteq> 0"
abs_eq: fixes xb :: "(char list \<times> 'a) list"   and xa :: "char list"   and x :: "'a" shows "update (St xb) xa x = St ((xa, x) # xb)"
Union_Pow_eq: fixes A :: "'b set" shows "\<Union> (Pow A) = A"
Amin_ge1Tr: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and z :: "ant" shows "(\<forall>j\<le>Suc n. f j \<in> Z\<^sub>\<infinity> \<and> z \<le> f j) \<longrightarrow> z \<le> Amin (Suc n) f"
wtl_def: fixes ins :: "'b list"   and cert :: "'a list" shows "wtl ins cert \<equiv> wtl_inst_list ins cert f r \<top> \<bottom> step"
omega_subdist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<omega> \<le> (x + y)\<^sup>\<omega>"
image_compose_permutations_right: fixes q :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "q permutes S" shows "{p \<circ> q |p. p permutes S} = {p. p permutes S}"
prob_component_neq_zero: fixes p :: "real"   and w :: "bool stream"   and n :: "nat" assumes "0 < p"   and "p < 1" shows "prob_component p w n \<noteq> 0"
normal_qe_pres\<^sub>1_o: fixes as :: "atom list" assumes "\<forall>a\<in>set as. hd_coeff a \<noteq> 0 \<and> divisor a \<noteq> 0" shows "Z.normal ((qe_pres\<^sub>1 \<circ> hd_coeffs1) as)"
triangle: 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 f :: "'b"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "src g = trg f" shows "V (H g (bicategory.lunit V H \<a> \<i> src trg f)) (\<a> g (src g) f) = H (bicategory.runit V H \<a> \<i> src g) f"
\<theta>_fv': fixes t :: "('b, 'a) Term.term" shows "fv (t \<cdot> \<theta>) = {}"
supp_Pair: fixes x :: "'d"   and y :: "'e" shows "supp (x, y) = supp x \<union> supp y"
asymp_equiv_add_left: fixes h :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "h \<in> o[F](g)" shows "((\<lambda>x. h x + f x) \<sim>[F] g) = (f \<sim>[F] g)"
\<theta>_def: shows "\<theta> = 2 * arccos (sqrt (real (N - M) / real N))"
vsv_vdomain_vsingleton_vrange_vsingleton: fixes a :: "V" assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = ZFC_in_HOL.set {a}" obtains b :: "V" where "\<R>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = ZFC_in_HOL.set {b}"
height_swapLeaves: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "height (swapLeaves t w\<^sub>a a w\<^sub>b b) = height t"
comp_ide_self: fixes a :: "'s" assumes "S.ide a" shows "a \<cdot> a = a"
satisfiableAppend_2: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" assumes "satisfiable (formula1 @ formula2)" shows "satisfiable formula2"
simps_2: fixes n :: "nat"   and n' :: "nat" shows "wellformed' n (Bound n') = (n' < n)"
override_on_apply_in: fixes a :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "a \<in> A" shows "(f ++\<^bsub>A\<^esub> g) a = g a"
path_entry_last: fixes p :: "nat list"   and n :: "nat" assumes "path_entry (g_E G) p n"   and "p \<noteq> []" shows "last p = 0"
pure_sep_conj_conj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(P \<and>* Q) h"   and "pure P"   and "pure Q" shows "P h \<and> Q h"
divide_gt1: fixes b :: "'a"   and c :: "rat"   and a :: "'a" assumes "b < c *R a"   and "c < 0" shows "a < (1 / c) *R b"
iffb_inv2_inv: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "(fb\<^sub>\<F> \<circ> fb\<^sup>-\<^sub>\<F>) \<phi> = \<phi>" shows "Inf_pres \<phi>"
2: shows "v * top \<sqinter> g = v * v\<^sup>T \<sqinter> g"
card_prod: shows "CARD('a \<times> 'b) = CARD('a) * CARD('b)"
diff_add_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "c + b - a = c + (b - a)"
distinct_map: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
has_vector_derivative_weaken: fixes f :: "real \<Rightarrow> 'b"   and D :: "'b"   and x :: "real"   and T :: "real set"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "(f has_vector_derivative D) (at x within T)"   and "x \<in> S"   and "S \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(g has_vector_derivative D) (at x within S)"
transpose_row_code: fixes A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and i :: "'d" shows "($h) (transpose_row A i) = (\<lambda>j. A $h j $h i)"
Cons_10: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
fdg: shows "(f\<^sub>1, f\<^sub>2, g') = splitFace g u v f vs"
ConjI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<psi> in v]" shows "[\<phi> \<^bold>& \<psi> in v]"
P_v0: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "lhd P = v0"
distinct_1: fixes x1 :: "char list \<Rightarrow> 'b option" shows "Env x1 \<noteq> Malformed"
deg_minus_eq: fixes p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int" assumes "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q \<noteq> deg R p" shows "deg R (p \<ominus>\<^bsub>UP R\<^esub> q) = max (deg R p) (deg R q)"
Let: fixes p :: "pat"   and t :: "trm"   and \<theta> :: "(name \<times> trm) list"   and u :: "trm" assumes "supp p \<sharp>* t"   and "distinct (pat_vars p)"   and "\<turnstile> p \<rhd> t \<Rightarrow> \<theta>" shows "(LET p = t IN u) \<longmapsto> \<theta>\<lparr>u\<rparr>"
analz_hash_nonce: shows "analz {M. \<exists>N. M = Hash (Nonce N)} = {M. \<exists>N. M = Hash (Nonce N)}"
generate_m_inv_closed: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and h :: "'e" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "inv\<^bsub>G\<^esub> h \<in> generate G H"
isNSCont_NSLIM_iff: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b" shows "isNSCont f a = f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a"
prv_cnjL_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (cnj \<phi> fls) fls)"
satisfies_eq_def: fixes v :: "nat \<Rightarrow> 'b"   and eq :: "nat \<times> linear_poly" shows "v \<Turnstile>\<^sub>e eq \<equiv> v (lhs eq) = rhs eq \<lbrace> v \<rbrace>"
get_ancestors_also_parent: fixes h :: "(_) heap"   and some_ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_ancestors some_ptr \<rightarrow>\<^sub>r ancestors"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set ancestors"   and "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h" shows "parent \<in> set ancestors"
abs_zero: shows "\<bar>0::'a\<bar> = (0::'a)"
closed_map_inclusion_eq: fixes X :: "'c topology"   and S :: "'c set" shows "closed_map (subtopology X S) X id = closedin X (topspace X \<inter> S)"
homfg: shows "homotopic_with_canon (\<lambda>z. True) (sphere (0::'a) 1 \<inter> S) (T - {0::'a}) f g"
rep_eq: fixes x :: "'b fpxs"   and xa :: "rat" shows "fpxs_nth (fpxs_compose_power x xa) = (\<lambda>xb. if 0 < xa then fpxs_nth x (xb / xa) else (0::'b))"
infinite_Ico_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less" shows "infinite (ord.atLeastLessThan less_eq less a b) = less a b"
encode_interfering_operator_exclusion_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "encode_interfering_operator_exclusion \<Pi> t \<equiv> let ops = \<Pi>\<^sub>\<O>; interfering = filter (\<lambda>(op\<^sub>1, op\<^sub>2). index ops op\<^sub>1 \<noteq> index ops op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2) (List.product ops ops) in foldr (\<^bold>\<and>) (concat (map (\<lambda>(op\<^sub>1, op\<^sub>2). map (\<lambda>k. encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) [0..<t]) interfering)) (\<^bold>\<not> \<bottom>)"
finite_reach_af_G_opt: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "finite (reach \<Sigma> \<up>af\<^sub>G\<^sub>\<UU> (Abs \<phi>))"
execn_block_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and bdy :: "('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate"   and p :: "'d" assumes "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> t" obtains t' :: "'e" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Normal t'__" and "\<Gamma>\<turnstile> \<langle>c s t'__,Normal (return s t'__)\<rangle> =n\<Rightarrow> t"   | t' :: "'e" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Abrupt t'" and "t = Abrupt (return s t')"   | f :: "'f" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Fault f" and "t = Fault f"   | "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Stuck" and "t = Stuck"   | "\<Gamma> p = None" and "t = Stuck"
dvd_div_mult_self: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "b div a * a = b"
assms_1: shows "f \<in> M \<rightarrow>\<^sub>M stock_measure t"
eventually_nhds_metric: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "eventually P (nhds a) = (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
serial_sas_plus_equivalent_to_serial_strips_i: fixes ops :: "('c \<times> 'd) strips_operator list"   and \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list" assumes "ops \<in> set (\<phi>\<^sub>P \<Psi> List_Supplement.embed \<psi>)" obtains op :: "('c, 'd) sas_plus_operator" where "op \<in> set \<psi>" and "ops = [\<phi>\<^sub>O \<Psi> op]"
le_less_linear: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
sep_conj_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P \<and>* Q \<equiv> \<lambda>h. \<exists>x y. x ## y \<and> h = x + y \<and> P x \<and> Q y"
intros_2: fixes n :: "nat" shows "[\<dots>n] \<in> i_set_ind"
adds_triv_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus a b)"
MTimesL_def: fixes r :: "mregex"   and S :: "mregex set" shows "MTimesL r S = MTimes r ` S"
distinct_3: fixes x11 :: "nat set"   and x12 :: "'d"   and x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d" shows "x11:*cast(x12) \<noteq> x31\<not>x32:arrive(x33)"
valid_cfgD: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg" assumes "Finite_Markov_Decision_Process K S"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S" shows "cfg \<in> Markov_Decision_Process.cfg_on K (state cfg)"
wf_digraph_axioms: shows "wf_digraph (with_proj (mk_symmetric H))"