hcju/isabelleps huggingface dataset:
qe_eq_repeat_helper_eval_case2: fixes xs :: "real list"   and var :: "nat"   and A :: "real mpoly list"   and L :: "atom list"   and F :: "atom fm list"   and x :: "real"   and \<Gamma> :: "real list" assumes "length xs = var"   and "\<forall>p\<in>set A. MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"   and "eval (qe_eq_repeat_helper var A L F) (xs @ x # \<Gamma>)" shows "\<exists>x. eval (list_conj (map (fm.Atom \<circ> Eq) A @ map fm.Atom L @ F)) (xs @ x # \<Gamma>)"
is_applicable_then_enabled_3: fixes s :: "nat \<Rightarrow> nat option"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "s \<in> valid_states"   and "lookup_operator ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) ast_op) = Some (name, pres, effs, layer)" shows "s \<in> subsuming_states (map_of (implicit_pres effs))"
f': fixes k :: "nat" shows "f' k = (if k = j then 0\<^sub>m d d else f k)"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "sup (b::'b) (sup (a::'b) (c::'b)) = sup a (sup b c)"
p_nsubs: shows "\<forall>D\<in>fst ` set P. \<not> strictly_subsume [C] D"
fmap_ext: fixes m :: "'c \<rightharpoonup>\<^sub>f 'd"   and n :: "'c \<rightharpoonup>\<^sub>f 'd" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
scan_take: fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and a :: "'d" shows "take k (scan f xs a) = scan f (take k xs) a"
uniformity_metric_completion_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
assms_5: shows "\<not> S1 \<sqsubseteq> S1 \<triangle> S2"
set_filter: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "set (filter P xs) = {x \<in> set xs. P x}"
epi_def: fixes f :: "'s" shows "epi f = (arr f \<and> inj_on (\<lambda>g. g \<cdot> f) {g. seq g f})"
ivl_disj_int_one_4: fixes l :: "'c"   and u :: "'c" shows "{..<l} \<inter> {l..u} = {}"
setsum_negf_ereal: fixes f :: "'a \<Rightarrow> ereal"   and I :: "'a set" assumes "\<not> {- \<infinity>, \<infinity>} \<subseteq> f ` I" shows "(\<Sum>i\<in>I. - f i) = - sum f I"
normDH1_2: shows "poly_mod.Mp p H1 = H1"
hyp: shows "\<forall>s. s \<in> set signs \<and> s \<notin> consistent_sign_vec qs ` set (characterize_root_list_p p) \<longrightarrow> z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}) = 0"
split_list: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> ran\<^sub>l xs" shows "\<exists>ys zs. xs = ys @ x # zs"
mod_add_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) (modulo b c)) c = modulo (plus a b) c"
assms: shows "\<tau> \<Turnstile> \<upsilon> foldl UML_Set.OclIncluding Set{} (map (f p) s_set)"
b_least2_scheme2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec3"   and "g \<in> PrimRec2"   and "\<forall>x y. h x y < g x y"   and "\<forall>x y. f x y (h x y) \<noteq> 0"   and "\<forall>z x y. z < h x y \<longrightarrow> f x y z = 0" shows "h \<in> PrimRec2"
deriv_ctxt: shows "\<forall>(P, p, Q, A)\<in>\<Theta>'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
dom_locals_fvar_vvar_mono: fixes s :: "state"   and statDeclC :: "qtname"   and stat :: "bool"   and fn :: "vname"   and a :: "val"   and s' :: "state"   and val :: "val" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (snd (fst (fvar statDeclC stat fn a s')) val s)))"
B: shows "cone {row A i |i. i < nr} = polyhedral_cone B"
fempty_fsubsetI: fixes x :: "'c fset" shows "{||} |\<subseteq>| x"
assms_1: shows "A = S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S'"
r_shrink_def: fixes f :: "recf" shows "r_shrink f \<equiv> Cn 1 f [recf.Id 1 0, recf.Id 1 0]"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le>\<cdot> y"   and "y \<le>\<cdot> z" shows "x \<le>\<cdot> z"
intvs_decr_l: fixes l :: "int"   and h :: "int" assumes "l \<le> h" shows "{l - 1..<h} = Set.insert (l - 1) {l..<h}"
measure_linear_sufficient: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and m :: "real" assumes "linear f"   and "S \<in> lmeasurable"   and "\<And>a b. Sigma_Algebra.measure lebesgue (f ` cbox a b) = m * Sigma_Algebra.measure lebesgue (cbox a b)" shows "f ` S \<in> lmeasurable \<and> m * Sigma_Algebra.measure lebesgue S = Sigma_Algebra.measure lebesgue (f ` S)"
fls_regpart_const: fixes c :: "'a" shows "fls_regpart (fls_const c) = fps_const c"
obs_upred_refine_iff: fixes Q :: "'a upred"   and P :: "'a upred" shows "(P \<sqsubseteq> Q) = (\<lbrakk>Q\<rbrakk>\<^sub>o \<subseteq> \<lbrakk>P\<rbrakk>\<^sub>o)"
mcont2mcont_lmap: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> 'b llist"   and f :: "'b \<Rightarrow> 'c" assumes "mcont lub ord lSup lprefix t" shows "mcont lub ord lSup lprefix (\<lambda>x. lmap f (t x))"
assms_3: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) C = Cv"
cauchy_in\<^sub>N_def: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c" shows "cauchy_in\<^sub>N N u = (\<forall>e>0. \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. Norm N (u n - u m) < e)"
conjugate_rat_def: fixes x :: "rat" shows "conjugate x \<equiv> x"
topcontinuous_at_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "topcontinuous_at X Y f x = (x \<in> topspace X \<and> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>V. openin Y V \<and> f x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y\<in>U. f y \<in> V))))"
simps_3: fixes c1 :: "'b acom"   and c2 :: "'b acom"   and c1' :: "'b acom"   and c2' :: "'b acom" shows "(c1;; c2 \<sqsubseteq> c1';; c2') = (c1 \<sqsubseteq> c1' \<and> c2 \<sqsubseteq> c2')"
not_neg_one_le_neg_numeral_iff: fixes 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(\<not> less_eq (uminus one) (uminus (numeral.numeral one plus m))) = (m \<noteq> num.One)"
cobounded2: fixes b :: "'a multiset"   and a :: "'a multiset" shows "b \<subseteq># a \<union># b"
Int_interior_Union_intervals: fixes \<F> :: "'a set set"   and S :: "'a set" assumes "finite \<F>"   and "open S"   and "\<And>T. T \<in> \<F> \<Longrightarrow> \<exists>a b. T = cbox a b"   and "\<And>T. T \<in> \<F> \<Longrightarrow> S \<inter> interior T = {}" shows "S \<inter> interior (\<Union> \<F>) = {}"
exhaust: fixes y :: "mode" obtains "y = I"   | "y = P"   | "y = V"
R_sum: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> R" shows "sum f A \<in> R"
hom: shows "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id"
alpha_Tree_rel_relcomp_trivialI: fixes x :: "('b, 'c, 'd, 'e) Tree"   and y :: "'f"   and R :: "(('b, 'c, 'd, 'e) Tree \<times> 'f) set" assumes "(x, y) \<in> R" shows "(x, y) \<in> FL_Validity.alpha_Tree_rel O R"
resCap_gzero_aux: fixes p :: "(nat \<times> nat) list" assumes "cf.isPath s p t" shows "(0::'capacity) < resCap p"
nonempty_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> \<noteq> {}"
subst_del_def: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and x :: "'c \<Longrightarrow> 'b" shows "\<sigma> -\<^sub>s x = \<sigma>(x \<mapsto>\<^sub>s &x)"
vsubstAbs_compose1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and y2 :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "goodAbs A" shows "A $[y1 // y]_ys $[y2 // y]_ys = A $[y1 @ys[y2 / y]_ys // y]_ys"
eq_sym: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "\<psi> \<sim>\<^sub>C \<phi>"
order_neg: fixes x :: "complex" shows "order x (charpoly cA) = order (x / cis (arg \<alpha>)) (charpoly cA)"
holomorphic_Dirichlet_L_weak: assumes "0 < n"   and "1 \<notin> A" shows "Dirichlet_L n (principal_dchar n) holomorphic_on A"
div_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
lastTrailElementMarkedImpliesMarkedElementsToLastElementAreAllMarkedElements: fixes M :: "('b \<times> bool) list" assumes "isDecision (last M)"   and "last (elements M) \<notin> set (butlast (elements M))" shows "markedElementsTo (last (elements M)) M = markedElements M"
l11_61: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and A' :: "'a"   and B :: "'a"   and C :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> A'"   and "A \<noteq> B"   and "A \<noteq> C"   and "Tarski_neutral_dimensionless.Coplanar Bet A A' B B'"   and "Tarski_neutral_dimensionless.Per Bet Cong B A A'"   and "Tarski_neutral_dimensionless.Per Bet Cong B' A' A"   and "Tarski_neutral_dimensionless.Coplanar Bet A A' C C'"   and "Tarski_neutral_dimensionless.Per Bet Cong C A A'"   and "Tarski_neutral_dimensionless.Per Bet Cong B A C" shows "Tarski_neutral_dimensionless.Per Bet Cong B' A' C'"
zmset_of_inverse: fixes M :: "'a multiset" shows "mset_pos (zmset_of M) = M"
spr_tLast: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t = spr_jview a t'" shows "envObs a (tLast t) = envObs a (tLast t')"
other_steps_resp_local: fixes A :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and I :: "'e set" shows "other_steps (other A I) I"
zero_le_scaleR_iff: fixes a :: "real"   and b :: "'a" shows "((0::'a) \<le> a *\<^sub>R b) = (0 < a \<and> (0::'a) \<le> b \<or> a < 0 \<and> b \<le> (0::'a) \<or> a = 0)"
Texp_IfThenElse: fixes scrut :: "exp"   and e1 :: "exp"   and a :: "Arity"   and e2 :: "exp" shows "Texp scrut\<cdot>0 \<otimes>\<otimes> (Texp e1\<cdot>a \<oplus>\<oplus> Texp e2\<cdot>a) \<sqsubseteq> Texp (scrut ? e1 : e2)\<cdot>a"
mn: shows "m - n = word_of_nat (mq - 2 ^ sq * nq) * 2 ^ s1"
sublist_every: fixes l1 :: "'c list"   and l2 :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "subseq l1 l2 \<and> list_all P l2" shows "list_all P l1"
division_of_union_self: fixes p :: "'c set set"   and s :: "'c set" assumes "p division_of s" shows "p division_of \<Union> p"
SigmaD2: fixes a :: "'c"   and b :: "'d"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
le_rem_s_def: fixes e :: "'c el_step" shows "le_rem_s e \<equiv> case e of LOC a \<Rightarrow> a | ENV a \<Rightarrow> a"
lemma_2_10_33: fixes b :: "'a"   and a :: "'a" shows "(((b r\<rightarrow> a) l\<rightarrow> a) r\<rightarrow> b) r\<rightarrow> b r\<rightarrow> a = b r\<rightarrow> a"
cnt_def: shows "cnt = (\<lambda>i. length (filter (\<lambda>y. (y, xs ! i) \<in> R) (delete_index i xs)))"
ccINF_lower2: fixes A :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'a"   and u :: "'a" assumes "countable A"   and "i \<in> A"   and "f i \<le> u" shows "Inf (f ` A) \<le> u"
has_contour_integral_linepath_same_Im_iff: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex"   and I :: "complex" assumes "Im a = Im b"   and "Re a < Re b" shows "(f has_contour_integral I) (linepath a b) = ((\<lambda>x. f (complex_of_real x + complex_of_real (Im a) * \<i>)) has_integral I) {Re a..Re b}"
EASSERT_def: fixes \<Phi> :: "bool" shows "EASSERT \<Phi> \<equiv> if \<Phi> then ERETURN () else FAIL"
00: shows "FvarsT t \<inter> snd ` set txs = {}"
value_unlift: fixes f :: "'v node \<times> 'v node \<Rightarrow> ennreal" assumes "flow \<Delta>' f" shows "value_flow \<Delta> (\<lambda>(x, y). f (Inner x, Inner y)) = value_flow \<Delta>' f"
subset: fixes R :: "('b, 'c) ring_scheme"   and N :: "'d set"   and M :: "('b, 'd, 'e) module_scheme" assumes "LinearCombinations.submodule R N M" shows "N \<subseteq> carrier M"
factorcount_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" shows "factorcount G a = (THE c. \<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as)"
cases: fixes a1 :: "'b list"   and a2 :: "'c sequent" assumes "(a1, a2) \<in> Ax" obtains i :: "nat" where "a1 = []" and "a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> otway"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> otway; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> otway; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 A B NA. \<lbrakk>evs1 \<in> otway; P evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> # evs1)"   and "\<And>evs2 B A NA NB. \<lbrakk>evs2 \<in> otway; P evs2; Gets B \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Agent A, Agent B, Nonce NA, Nonce NB\<rbrace> # evs2)"   and "\<And>evs3 KAB A B NA NB. \<lbrakk>evs3 \<in> otway; P evs3; Key KAB \<notin> used evs3; Gets Server \<lbrace>Agent A, Agent B, Nonce NA, Nonce NB\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server B \<lbrace>Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B, Key KAB\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Agent A, Agent B, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 B A NA NB X K. \<lbrakk>evs4 \<in> otway; P evs4; B \<noteq> Server; Says B Server \<lbrace>Agent A, Agent B, Nonce NA, Nonce NB\<rbrace> \<in> set evs4; Gets B \<lbrace>X, Crypt (shrK B) \<lbrace>Nonce NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A X # evs4)"   and "\<And>evso B A NA K NB. \<lbrakk>evso \<in> otway; P evso; Says Server B \<lbrace>Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
insertI: fixes A :: "'c set"   and a :: "'c" assumes "finite A" shows "finite (insert a A)"
mult_le_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times c a) (times c b) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
gt_def: fixes c :: "int"   and t :: "Ferrack.num" shows "gt c t = (if c = 0 then Gt t else if 0 < c then Gt (CN 0 c t) else Lt (CN 0 (- c) (Neg t)))"
congBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s\<^sub>e Q" shows "P \<sim>\<^sub>e Q"
preserving: fixes u :: "complex_homo"   and \<phi> :: "real" assumes "u \<in> unit_disc"   and "u \<noteq> 0\<^sub>h"   and "P (moebius_pt (moebius_rotation \<phi>) u)" shows "P u"
34: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "(x \<squnion> y) \<sqinter> z \<squnion> u = x \<sqinter> z \<squnion> (y \<sqinter> z \<squnion> u)"
BVG_loop: fixes \<alpha> :: "game" shows "BVG \<alpha>** \<subseteq> BVG \<alpha>"
rel_F_neg_distr_cond_eq: fixes tytok :: "('v \<times> 'w \<times> 'x \<times> 'y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae) itself" shows "rel_F_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"
F2set3_IF2set: fixes a :: "'d IF2"   and x :: "('d, 'd IF1, 'd IF2) F2" assumes "a \<in> F2set3 x" shows "IF2set a \<subseteq> IF2set (ctor2 x)"
bounded_iff: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a \<le> min b c) = (a \<le> b \<and> a \<le> c)"
variance: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "\<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs \<or> nxt \<phi> xs) xs" shows "alw \<phi> xs \<or> ev \<psi> xs"
unit_factor_1: shows "unit_factor (1::'a) = (1::'a)"
refl: fixes g :: "'b"   and succs :: "'b \<Rightarrow> 'b list" shows "g [succs]\<rightarrow>* g"
intros: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
M_rel_iD_1: fixes f1 :: "'i"   and f3 :: "int" assumes "M_rel_i f1 f3" shows "R f1 (of_int (M f3))"
path_append_elim: fixes r :: "'action list"   and s :: "'action list"   and p :: "'state" assumes "path (r @ s) p" obtains "path r p" and "path s (target r p)"
less_SUP_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
measurable_count_space_eq2_countable: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and A :: "'d set" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
is_basis_def: fixes V :: "'b set"   and B :: "'b set" shows "is_basis V B = (B \<subseteq> V \<and> local.independent B \<and> V \<subseteq> local.span B \<and> card B = local.dim V)"
empty_subst_spec: shows "(\<forall>f. subst empty_subst f = f) \<and> subst_lconsts empty_subst = {}"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>* (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = f__" and "a3 = ms__" and "a4 = sa__" and "a5 = []" and "a6 = ms__" and "a7 = sa__" and "length ms__ = length sa__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as' :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = last as # as'" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" and "S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s'')"
simps_2: fixes y :: "'a"   and ys :: "'a list" shows "proper_interval_set_aux [] (y # ys) = (ys \<noteq> [] \<or> proper_interval (Some y) None)"
is_strict_rewrite_ordD: fixes rel :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" assumes "is_strict_rewrite_ord rel" shows "is_rewrite_ord (\<lambda>x y. \<not> rel y x)"
UN_simps_10: fixes B :: "'d \<Rightarrow> 'c set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "\<Union> (B ` f ` A) = (\<Union>a\<in>A. B (f a))"
rconsNlist_inv: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and xs :: "('c, 'd) Node list" assumes "is_leveln_ftree n t"   and "is_measured_ftree t"   and "\<forall>x\<in>set xs. is_leveln_node n x \<and> is_measured_node x" shows "is_leveln_ftree n (rconsNlist t xs) \<and> is_measured_ftree (rconsNlist t xs)"
full_SetCompr_eq: fixes f :: "'f \<Rightarrow> 'e" shows "{u. \<exists>x. u = f x} = range f"
subtype_OclSuper_x: fixes \<sigma> :: "'a OCL_Types.type"   and P :: "bool" assumes "OclSuper \<sqsubset> \<sigma>" shows "P"
assms_2: shows "\<lbrace>x\<rbrace>y\<lbrace>z\<rbrace>"
colossless_gpv_continuationD: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv"   and input :: "'b" assumes "colossless_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "colossless_gpv \<I> (c input)"
opt: shows "fst (Partial_Cost_Model.config' (\<lambda>s. [], TS_step_d) (Lxy init {x, y}, []) (bs @ [a] @ f)) \<in> {[a, e], [e, a]}"
disc_rfr_proc_borel_measurable: fixes r :: "real"   and n :: "nat"   and M :: "'c measure" shows "disc_rfr_proc r n \<in> borel_measurable M"
HPhase1or2Write_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and v :: "InputsOrNi" assumes "HPhase1or2Write s s' q d"   and "valueChosen s v"   and "v \<in> Inputs"   and "HInv4a s q" shows "valueChosen s' v"
assms_3: assumes "e' = e\<^sub>2[e\<^sub>1 / x] \<and> value e\<^sub>1 \<and> \<pi> = \<pi>'" shows "Q"
code: fixes r :: "'a language"   and s :: "'a language" shows "Plus r s = Lang (\<oo> r \<or> \<oo> s) (\<lambda>a. Plus (\<dd> r a) (\<dd> s a))"
return_rule: fixes x :: "'c" shows "<emp> return x <\<lambda>r. \<up> (r = x)>"
ord_spmf_reflI: fixes p :: "'d spmf"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord x x" shows "ord_spmf ord p p"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>hp. P \<langle>\<rangle> hp"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>hsx x hsy y. P \<langle>hsx, x, \<langle>\<rangle>\<rangle> \<langle>hsy, y, \<langle>\<rangle>\<rangle>"   and "\<And>v va vf vg vh vc vd ve. P \<langle>v, va, \<langle>vf, vg, vh\<rangle>\<rangle> \<langle>vc, vd, ve\<rangle>"   and "\<And>v va vb vc vd vf vg vh. P \<langle>v, va, vb\<rangle> \<langle>vc, vd, \<langle>vf, vg, vh\<rangle>\<rangle>" shows "P a0 a1"
star_if_lang: fixes w :: "'c list"   and A :: "'c list set" assumes "w \<in> A" shows "w \<in> star A"
2: shows "\<forall>a. ListMem a as \<longrightarrow> agree (fmrestrict_set vs (snd a)) (fmrestrict_set vs s)"