hcju/isabelleps huggingface dataset:
upt_eq_append_conv: fixes i :: "nat"   and j :: "nat"   and xs :: "nat list"   and ys :: "nat list" assumes "i \<le> j" shows "([i..<j] = xs @ ys) = (\<exists>k\<ge>i. k \<le> j \<and> [i..<k] = xs \<and> [k..<j] = ys)"
set_insort_key: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" shows "set (insort_key f x xs) = insert x (set xs)"
tauLaw4CongSimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<leadsto>\<guillemotleft>Rel\<guillemotright> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
rep_eq: fixes x :: "uint8"   and xa :: "nat" shows "Rep_uint8 (x >>> xa) = signed_drop_bit xa (Rep_uint8 x)"
smc_dg_SemiCAT: fixes \<alpha> :: "V" shows "smc_dg (smc_SemiCAT \<alpha>) = dg_SemiCAT \<alpha>"
list_all2_mono: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys" shows "list_all2 Q xs ys"
strict_not_outside_2: fixes x :: "'b"   and y :: "'b" assumes "x \<prec>[Pareto R'] y" shows "y \<in> alts'"
append_self_conv2: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<cdot> ys = ys) = (xs = \<epsilon>)"
abs_0: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs zero = zero"
gH_rel_def: fixes x :: "'a"   and y :: "'a" shows "(x \<preceq>\<^sub>H y) = (x \<preceq>\<^sub>L y \<and> x \<preceq>\<^sub>R y)"
gcd_dvd1: fixes a :: "'b"   and b :: "'b" shows "some_gcd a b dvd a"
scalar_prod_def: fixes v :: "'b vec"   and w :: "'b vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $ i * w $ i"
document_ptr_kinds_eq2: shows "|h \<turnstile> document_ptr_kinds_M|\<^sub>r = |h' \<turnstile> document_ptr_kinds_M|\<^sub>r"
comp_assoc': fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr"   and h :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f"   and "J.seq h g" shows "(h \<cdot>\<^sub>J g) \<cdot>\<^sub>J f = h \<cdot>\<^sub>J g \<cdot>\<^sub>J f"
stmt: fixes m :: "'a \<times> 'a set" assumes "m \<in># {#a#} \<times># N" shows "(fst m \<in> snd m) = (a \<in> snd m)"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> 'i" assumes "\<not> P5 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "'i" where "x = (s1, s2)" and "\<not> (\<forall>p. status s1 p \<noteq> Sleep \<and> initialized s1 \<and> initVals s1 = {})"
disc_1: fixes x1 :: "'f" shows "is_Var (Var x1)"
f': fixes e :: "'v \<times> 'v" shows "(\<lambda>i. f_i' i e) \<longlonglongrightarrow> f' e"
witness_2: fixes a :: "'b" assumes "a \<in> S" shows "j a \<in> T"
iprev_in_iff: fixes n :: "nat"   and I :: "nat set" shows "(iprev n I \<in> I) = (n \<in> I)"
distinct_5: fixes x21 :: "'a"   and x22 :: "'a form list" shows "Compound x21 x22 \<noteq> ff"
coprime_mod_right_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "a \<noteq> zero" shows "algebraic_semidom.coprime times one a (modulo b a) = algebraic_semidom.coprime times one a b"
Osum_wf: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "wf r"   and "wf r'" shows "wf (r \<union>o r')"
fvi_iff_fv: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "Formula.formula" shows "(x \<in> Formula.fvi b \<phi>) = (x + b \<in> fv \<phi>)"
arr_char': fixes f :: "(bool, 'arr list) concrete_category.arr" shows "arr f = (f = MkIde False \<or> f = MkIde True \<or> f \<in> (\<lambda>x. MkArr False True [x]) ` Arr)"
distinct_5: fixes x1 :: "agent"   and x4 :: "nat" shows "Agent x1 \<noteq> Pan x4"
span_Times_sing2: fixes A :: "'b set" shows "p.span (A \<times> {0::'c}) = vs1.span A \<times> {0::'c}"
is_alph_def: fixes n :: "nat" shows "is_alph n = (\<lambda>w. length w = n)"
absc_valid: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "absc cfg \<in> R_G.valid_cfg"
inf_sup_aci_6: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "sup_prod (sup_prod x y) z = sup_prod x (sup_prod y z)"
AA15: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<langle>P \<and> Q\<rangle>_v \<longrightarrow> \<diamond>\<langle>P\<rangle>_v"
assms: shows "\<forall>x\<in>set xs. real (length (f x)) \<le> g x"
segment_x_x_eq: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "permutation f" shows "segment f x x = orbit f x - {x}"
WHILEI_invisible_refine_genR: fixes I' :: "'c \<Rightarrow> bool"   and s' :: "'c"   and s :: "'d"   and R' :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres"   and R :: "('d \<times> 'c) set" assumes "I' s' \<Longrightarrow> (s, s') \<in> R'"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R'; I' s'\<rbrakk> \<Longrightarrow> I s"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R'; I' s'; I s; b' s'\<rbrakk> \<Longrightarrow> b s"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R'; I' s'; I s; b s\<rbrakk> \<Longrightarrow> f s \<le> sup (\<Down> R' (ASSUME (b' s') \<bind> (\<lambda>_. f' s'))) (\<Down> R' (RETURN s'))"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; \<not> b x; \<not> b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> (x, x') \<in> R" shows "WHILE\<^bsup>I\<^esup> b f s \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' s')"
simps_8: fixes P :: "pi"   and Q :: "pi" shows "summands (P \<parallel> Q) = {}"
holo: fixes A :: "complex set" assumes "A \<subseteq> {s. 0 < Re s}" shows "zeta' holomorphic_on A"
norm_respects_equiv: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "local.equiv t u" shows "\<^bold>\<parallel>t\<^bold>\<parallel> = \<^bold>\<parallel>u\<^bold>\<parallel>"
assms: shows "low_equal s1 s2 \<and> Some t1 = store_word_mem s1 addr data bm 10 \<and> Some t2 = store_word_mem s2 addr data bm 10"
mult_fract: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "Fract a b * Fract c d = Fract (a * c) (b * d)"
bidirected_digraph_axioms: shows "bidirected_digraph H rev_H'"
earlyLateTau: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'" shows "P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'"
inv_integer_group: fixes x :: "int" shows "inv\<^bsub>integer_group\<^esub> x = - x"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism S S f g"   and "f u = v"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> ball a r \<inter> T" shows "thesis"
pringsheim_landau_aux: fixes f :: "complex fds"   and c :: "real"   and A :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "nonneg_dirichlet_series f"   and "abs_conv_abscissa f \<le> ereal c"   and "\<And>s. \<lbrakk>s \<in> A; c < Re s\<rbrakk> \<Longrightarrow> g s = eval_fds f s"   and "g holomorphic_on A"   and "open A"   and "complex_of_real c \<in> A" shows "\<exists>x<c. fds_abs_converges f (complex_of_real x)"
twoCisTypeII_9: shows "(\<lambda>x. twoC (1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
AXHp_AXH_eq: shows "AXHp = (\<lambda>x. x \<in> AXH)"
assms: shows "dep_var_set PROB (\<Union> S) vs"
assms_2: shows "B \<in>\<^sub>\<circ> Vset \<alpha>"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
cross7_simps_50: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "(a - b) * x = a * x - b * x"
that: fixes xs1 :: "'d list"   and xs2 :: "'d list" assumes "ys1 = map f xs1"   and "ys2 = map f xs2"   and "xs = xs1 @ xs2" shows "thesis"
rel_simps_43: shows "\<not> (1::'a) \<le> - (1::'a)"
CC: fixes F :: "'a formula"   and \<Gamma> :: "'a formula set" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> F"
comp_arr_inv': fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.comp f (CC.inv f) = CC.cod f"
replacefacesAt_def: fixes ns :: "nat list"   and f :: "face"   and fs :: "face list"   and F :: "face list list" shows "replacefacesAt ns f fs F \<equiv> mapAt ns (replace f fs) F"
P3_cong: fixes tms :: "tm list"   and P :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"   and H :: "fm set"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm"   and z :: "tm"   and z' :: "tm" assumes "\<And>i t u v x. atom i \<sharp> tms \<Longrightarrow> (P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"   and "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'"   and "H \<turnstile> z EQ z'" shows "H \<turnstile> P x y z IFF P x' y' z'"
assms_1: shows "exec_gpv1 \<equiv> exec_gpv"
path2_cases: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" obtains "ns = [n]" and "m = n"   | "g \<turnstile> hd (tl ns)-tl ns\<rightarrow>m" and "n \<in> set (old.predecessors g (hd (tl ns)))"
syncHdAddEmpty: fixes a :: "'a event list"   and A :: "'a event set"   and u :: "'a event list" assumes "tl a setinterleaves (([], u), A) \<and> hd a \<notin> A \<and> a \<noteq> []" shows "a setinterleaves (([], hd a # u), A)"
epi_iso_compose_rev: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme"   and g :: "'c \<Rightarrow> 'f"   and K :: "('f, 'g) monoid_scheme" assumes "f \<in> epi G H"   and "g \<in> hom H K"   and "g \<circ> f \<in> Group.iso G K" shows "f \<in> Group.iso G H \<and> g \<in> Group.iso H K"
div0_def: shows "div0 \<equiv> inv\<^bsub>class_ring\<^esub> (0::'b)"
hf: shows "Ftor (YFtor C @@ X) : Op C \<longrightarrow> SET"
f_integrable: shows "\<forall>twoCube\<in>twoChain. F integrable_on cubeImage twoCube"
act_conj_act_4: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
slide_hom: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "slide i \<in> {l. l \<le> j - i} \<rightarrow> nset i j"
ord_strict_p_transitive: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c \<Rightarrow>\<^sub>0 'f"   and b :: "'c \<Rightarrow>\<^sub>0 'f"   and c :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.ord_strict_p ord_term_strict a b"   and "ordered_term.ord_strict_p ord_term_strict b c" shows "ordered_term.ord_strict_p ord_term_strict a c"
lim_const_over_n: fixes a :: "'a" shows "(\<lambda>n. a / of_nat n) \<longlonglongrightarrow> (0::'a)"
the_adj_chamber_adjacentset: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "the_adj_chamber C z \<in> adjacentset C"
unstar_star_of: fixes p :: "bool" shows "unstar (star_of p) = p"
list_rel_pred_zip: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P xs ys = (length xs = length ys \<and> list_all (\<lambda>(x, y). P x y) (xs || ys))"
gt_state_def: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" shows "s \<succ>\<^sub>x s' \<equiv> \<exists>x\<^sub>i x\<^sub>j l\<^sub>i. let dir = if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative in gt_state' dir s s' x\<^sub>i x\<^sub>j l\<^sub>i"
Setcompr_eq_image: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set" shows "{f x |x. x \<in> A} = f ` A"
simps_2: fixes P :: "'a"   and Cal :: "'a processCalculus"   and n :: "nat"   and Q :: "'a" shows "P \<longmapsto>Cal\<^bsup>Suc n\<^esup> Q = (\<exists>P'. P \<longmapsto>Cal\<^bsup>n\<^esup> P' \<and> P' \<longmapsto>Cal Q)"
P9: shows "Bet R C' A \<and> Cong S C R C'"
fold_commute_apply: fixes xs :: "'g list"   and h :: "'h \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and f :: "'g \<Rightarrow> 'i \<Rightarrow> 'i"   and s :: "'h" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h (fold g xs s) = fold f xs (h s)"
segment_not_inc_iff: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (b \<notin> segment D a)"
inj_on_fm: fixes M :: "'i \<Rightarrow> 'e measure"   and n :: "nat" assumes "\<And>i. space (M i) = UNIV" shows "inj_on (fm n) (space (Pi\<^sub>M (J n) M))"
assms_2: fixes x :: "'a" assumes "x \<in> indets q" shows "f x = g x"
oalist_of_list_def: shows "oalist_of_list = OAlist"
nlcons_inv_2: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_measured_ftree t"   and "is_measured_node nd" shows "is_measured_ftree (nlcons nd t)"
prime_factors_nat_def: fixes n :: "nat" shows "prime_factors_nat n = remdups_adj (sort (prime_factorization_nat n))"
map_upd_Some_unfold: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a"   and x :: "'b"   and y :: "'a" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
R\<^sub>C_ofI: fixes c\<^sub>1\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>1\<^sub>C :: "'Com\<^sub>C"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>2\<^sub>A :: "'Com\<^sub>A"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C :: "'Com\<^sub>C"   and \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "(\<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"   and "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A) \<in> \<R>\<^sub>A"   and "conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C"   and "(\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P" shows "(\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> R\<^sub>C_of \<R>\<^sub>A \<R> P"
CondThrow: 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 e' :: "expr"   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 e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
s_id_par_s_prod: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<inter> 1\<^sub>\<sigma> \<parallel> (y \<inter> 1\<^sub>\<sigma>) = x \<inter> 1\<^sub>\<sigma> \<cdot> (y \<inter> 1\<^sub>\<sigma>)"
scaleC_right_mono: fixes a :: "complex"   and b :: "complex"   and x :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> x" shows "a *\<^sub>C x \<le> b *\<^sub>C x"
rp_bad_def: shows "rp_bad = (\<lambda>(bad, \<sigma>) x. case \<sigma> x of None \<Rightarrow> spmf_of_set A \<bind> (\<lambda>y. if y \<in> ran \<sigma> then map_spmf (\<lambda>y'. (y', True, \<sigma>(x \<mapsto> y'))) (spmf_of_set (A - ran \<sigma>)) else return_spmf (y, bad, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, bad, \<sigma>))"
select: shows "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"
sigma_finite_countable: fixes s :: "'s" shows "\<exists>A. countable A \<and> A \<subseteq> T.events s \<and> \<Union> A = space (T s) \<and> (\<forall>a\<in>A. emeasure (T s) a \<noteq> \<infinity>)"
gcsrp_top_closed: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> gcsrp_map G N ` set_rcs G N"   and "b \<in> gcsrp_map G N ` set_rcs G N" shows "gcsrp_top G N a b \<in> gcsrp_map G N ` set_rcs G N"
prod_alg_dvd_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x alg_dvd g x" shows "prod f A alg_dvd prod g A"
inputChainAlpha': fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "('d, 'e, 'f) psi"   and N :: "'d" assumes "(p \<bullet> xvec) \<sharp>* P"   and "(p \<bullet> xvec) \<sharp>* N"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" shows "inputChain xvec N P = inputChain (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P)"
K_imply_head: fixes p :: "'b fm"   and ps :: "'b fm list" shows "\<turnstile> imply (p # ps) p"
extended_hyperb_ineq: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and z :: "'a Gromov_completion" shows "min (extended_Gromov_product_at e x y) (extended_Gromov_product_at e y z) - ereal (deltaG TYPE('a)) \<le> extended_Gromov_product_at e x z"
hyps: shows "\<forall>y\<in>supp (P, Q, x). P[x::=y] \<equiv>\<^sub>e Q[x::=y] \<and> P[x::=y] \<sim> Q[x::=y]"
simps_3: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "document_element \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = document_element"
arr_char: fixes f :: "'c \<times> 'c" shows "C'.VV.arr f = (C.arr (fst f) \<and> C.arr (snd f) \<and> C'.src (fst f) = C'.trg (snd f))"
not_min_gallery_double_split: fixes Cs :: "'a set list" assumes "gallery Cs"   and "\<not> min_gallery Cs"   and "{} \<notin> set (wall_crossings Cs)" shows "\<exists>f g As A B Bs E F Fs. (f, g) \<in> foldpairs \<and> A \<in> f \<turnstile> \<C> \<and> B \<in> g \<turnstile> \<C> \<and> E \<in> g \<turnstile> \<C> \<and> F \<in> f \<turnstile> \<C> \<and> (Cs = As @ [A, B, F] @ Fs \<or> Cs = As @ [A, B] @ Bs @ [E, F] @ Fs)"
N_def: shows "\<forall>m>N. eint k < val_Zp_dist (s m) a"
indep_events_def_alt: fixes A :: "'d \<Rightarrow> 'a set"   and I :: "'d set" shows "indep_events A I = indep_sets (\<lambda>i. {A i}) I"
transI: fixes r :: "('a \<times> 'a) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
prems_3: shows "xs = ys @ j # concat (map (\<lambda>xs. xs @ [j]) (zs # xss)) @ as"
lappend_LNil2: fixes xs :: "'b llist" shows "xs @@ LNil = xs"
poslsts_def: fixes A :: "'b set" shows "A\<^sup>\<spadesuit> \<equiv> A\<^sup>\<infinity> - {LNil}"
SN_reaches_NF: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "SN_on r {x}" shows "\<exists>y. (x, y) \<in> r\<^sup>* \<and> y \<in> NF r"
weight_spmf_def: fixes p :: "'b spmf" shows "weight_spmf p = Sigma_Algebra.measure (measure_spmf p) UNIV"
computation_downpos_prelim: shows "rat_poly_inv (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B) = rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A"
sum_list_sets_single: fixes A :: "'a" shows "(\<Sum>X\<leftarrow>[A]. X) = A"
order_refl: fixes x :: "'b list" shows "x \<le>f x"
sin_double: fixes x :: "'a" shows "sin ((2::'a) * x) = (2::'a) * sin x * cos x"
1: shows "\<forall>m<n. insertion e (a ^\<^sub>p m) = insertion e a ^ m"
nonneg: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "1 \<le> x - ln x"
has_derivative_eq_rhs: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') F"   and "f' = g'" shows "(f has_derivative g') F"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<Z> \<alpha>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
integrable_mult_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x * c)"
vinsert_in_Vset_succI': fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)" shows "vinsert a b \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
coeff_replace_coeff: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "f (0::'b) = (0::'a)" shows "MPoly_Type.coeff (replace_coeff f p) m = f (MPoly_Type.coeff p m)"