hcju/isabelleps huggingface dataset:
val_Zp_0_criterion: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x 1 \<noteq> 0" shows "val_Zp x = 0"
functor_axioms: shows "functor C C (\<lambda>x. local.prod x b)"
trivial_bounds_cos: shows "\<forall>x. cos x \<in> {- 1..1}"
prod_eqI: fixes p :: "'c \<times> 'd"   and q :: "'c \<times> 'd" assumes "fst p = fst q"   and "snd p = snd q" shows "p = q"
simps_2: fixes p :: "nat"   and ea :: "floatarith"   and eas :: "floatarith list"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_slp p (ea # eas) xs = approx_floatarith p ea xs \<bind> (\<lambda>r. approx_slp p eas (r # xs))"
per_suppa__per: 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"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.Per Bet Cong D E F"
nn_integral_cong: fixes M :: "'k measure"   and u :: "'k \<Rightarrow> ennreal"   and v :: "'k \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
prems: shows "prefix B (A @ [a] \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
irreducible_imp_prime_elem: fixes x :: "'a" assumes "irreducible x" shows "prime_elem x"
smod_int_compares_4: fixes a :: "int"   and b :: "int" assumes "a \<le> 0"   and "0 < b" shows "a smod b \<le> 0"
bigomega_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "\<Omega>[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)}"
ceiling_altdef: fixes x :: "'d" shows "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)"
proportionality_refl_on: shows "refl_on local.non_zero_vectors local.proportionality"
isnpolyh_polynate_id: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "polynate p = p"
higher_differentiable_on_mult: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x * g x) n"
one_closed: shows "\<one>\<^bsub>A\<^esub> \<in> carrier A"
card_of_unique: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "card_order_on A r" shows "r =o |A|"
z_def: shows "z \<equiv> getFr [xx, yy, zz] [t] []"
f_def: shows "f = (SOME f. f \<in> ground_heads_var x)"
inv_equality: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and x :: "'c" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
min_union_empty_2: fixes A :: "'b fset set" shows "{} \<union>\<^sub>m A = min_set A"
prime_factorization_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize p = {#p#}"
Qg_iop_closed: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set"   and A :: "'c set" assumes "Group G"   and "G \<triangleright> N"   and "A \<in> set_rcs G N" shows "c_iop G N A \<in> set_rcs G N"
cnj_scaleR: fixes r :: "real"   and x :: "octo" shows "Octonions.cnj (r *\<^sub>R x) = r *\<^sub>R Octonions.cnj x"
2: shows "(- 1) ^ (card X - Suc j) = - ((- 1) ^ (card X - j))"
less_b_least2_impl_zero2: fixes z :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "z < y"   and "b_least2 f x y = y" shows "f x z = 0"
vector_eq: fixes x :: "'c"   and y :: "'c" shows "(x = y) = (x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x)"
swap_cols_rows_block_dims_main: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (swap_cols_rows_block i j A) = dim_row A \<and> dim_col (swap_cols_rows_block i j A) = dim_col A"
simplicial_subdivision_aux: fixes \<M> :: "'b set set"   and n :: "nat" assumes "finite \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"   and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> int n"   and "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"   and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1" shows "\<exists>\<T>. simplicial_complex \<T> \<and> (\<forall>K\<in>\<T>. aff_dim K \<le> int n) \<and> \<Union> \<T> = \<Union> \<M> \<and> (\<forall>C\<in>\<M>. \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F) \<and> (\<forall>K\<in>\<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
CX5h_range: fixes h :: "H2"   and X :: "X5 set" shows "CX5h h X \<subseteq> {x \<in> X. X5h x = h}"
lcoeff_nonzero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<noteq> \<zero>\<^bsub>P\<^esub>"   and "p \<in> carrier P" shows "coeff P p (deg R p) \<noteq> \<zero>"
mem_simps_11: fixes A :: "'a"   and C :: "'a set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
simps: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a4 :: "'Var \<Rightarrow> 'Val" shows "(a1, a2) \<rightarrow> (a3, a4) = (\<exists>cms n mem cm' mem'. a1 = cms \<and> a2 = mem \<and> a3 = cms[n := cm'] \<and> a4 = mem' \<and> (cms ! n, mem) \<leadsto> (cm', mem') \<and> n \<sqsubset> length cms)"
least_closed: fixes L :: "('c, 'd) gorder_scheme"   and l :: "'c"   and A :: "'c set" assumes "least L l A" shows "l \<in> carrier L"
vector_matrix_mult_def: fixes v :: "('a, 'c) vec"   and m :: "(('a, 'b) vec, 'c) vec" shows "v v* m \<equiv> \<chi>j. \<Sum>i\<in>UNIV. m $ i $ j * v $ i"
maddux2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "f x \<sqinter> y \<le> f (x \<sqinter> conjugate f y)"
xor_def: fixes A :: "bool"   and B :: "bool" shows "A \<oplus> B \<equiv> A \<and> \<not> B \<or> \<not> A \<and> B"
p_coprime_left_int: fixes a :: "int" shows "coprime (int p) a = (\<not> int p dvd a)"
Max_less_iMin_imp_disjoint: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "Max A < iMin B" shows "A \<inter> B = {}"
induct: fixes x1 :: "('a, 'b) ta_rule set"   and x2 :: "'b tree"   and x3 :: "'a"   and P :: "('a, 'b) ta_rule set \<Rightarrow> 'b tree \<Rightarrow> 'a \<Rightarrow> bool" assumes "accs_laz x1 x2 x3"   and "\<And>q f qs \<delta> ts. \<lbrakk>q \<rightarrow> f qs \<in> \<delta>; list_all_zip (\<lambda>x1 x2. accs_laz \<delta> x1 x2 \<and> P \<delta> x1 x2) ts qs\<rbrakk> \<Longrightarrow> P \<delta> (NODE f ts) q" shows "P x1 x2 x3"
mono_Seq_sym: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(S `;` P) \<sqsubseteq> (S `;` Q)"
fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_def: fixes F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list" shows "fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<equiv> concat (map (\<lambda>(t, t'). fv_list t @ fv_list t') F)"
numeral_BitM: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "numeral.numeral one plus (Num.BitM n) = minus (numeral.numeral one plus (num.Bit0 n)) one"
inter_mono: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "A \<subseteq> B" shows "A \<inter> C \<subseteq> B \<inter> C"
return_reads: fixes x :: "'n"   and h :: "'l"   and h' :: "'l" shows "reads {} (return x) h h'"
F_def: shows "F = {g. (\<forall>e. 0 \<le> g e) \<and> (\<forall>e. e \<notin> \<^bold>E \<longrightarrow> g e = 0) \<and> (\<forall>x. x \<notin> B \<Gamma> \<longrightarrow> d_IN g x \<le> d_OUT g x) \<and> linkage \<Gamma> g \<and> (\<forall>x\<in>A \<Gamma>. d_IN g x = 0) \<and> (\<forall>x. d_OUT g x \<le> weight \<Gamma> x) \<and> (\<forall>x. d_IN g x \<le> weight \<Gamma> x) \<and> (\<forall>x\<in>B \<Gamma>. d_OUT g x = 0) \<and> g \<le> f'}"
map_snd_def: fixes f :: "'b \<Rightarrow> 'c" shows "map_snd f = map (\<lambda>(x, y). (x, f y))"
ar_coset_def: fixes a :: "'c"   and A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" shows "a \<uplus>\<^bsub>A\<^esub> H = H \<bullet>\<^bsub>b_ag A\<^esub> a"
W2c: fixes b :: "'a"   and a :: "'a" shows "(b r\<rightarrow> a) l\<rightarrow> a = (a r\<rightarrow> b) l\<rightarrow> b"
post_def: shows "post = (\<lambda>(lo, hi, xs). quicksort_post R h lo hi xs)"
inner_Basis: fixes u :: "'a"   and v :: "'a" assumes "u \<in> Basis"   and "v \<in> Basis" shows "u \<bullet> v = (if u = v then 1 else 0)"
some_eq_ex: fixes P :: "'a \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
assoc_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "src \<a>[f, g, h] = src h"
mult_of_is_Units: shows "Multiplicative_Group.mult_of R = units_of R"
HPhase1or2ReadThen_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d r" shows "allBlocks s' \<subseteq> allBlocks s"
getMinTree_min_tree: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "t \<in> set bq" shows "SkewBinomialHeapStruc.prio (SkewBinomialHeapStruc.getMinTree bq) \<le> SkewBinomialHeapStruc.prio t"
lincomb_to_dvd_modulo: fixes u :: "int poly"   and c :: "nat \<Rightarrow> int"   and k :: "int"   and g :: "int vec" assumes "degree u \<le> n"   and "lincomb_list c (factorization_lattice u (n - degree u) k) = g" shows "poly_mod.dvdm k u (poly_of_vec g)"
encode_state_variable_is_lit_plus_if: fixes \<Pi> :: "'b strips_problem"   and v :: "'b"   and s :: "'b \<Rightarrow> bool option"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "v \<in> dom s" shows "is_lit_plus (encode_state_variable k (index (\<Pi>\<^sub>\<V>) v) (s v))"
R0: fixes X :: "'b list \<times> 'b list \<times> 'b list" shows "X \<in> l_sort_set X"
exe_complete_tcsigs_iff_complete_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss" shows "complete_tcsigs (set cs) (translate_ars arss) = exe_complete_tcsigs cs arss"
ind_eq_0_iff: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "(ind P n = (0::'b)) = (n = 0 \<or> \<not> P n)"
cont_both1: fixes y :: "'b ttree" shows "cont (\<lambda>x. x \<otimes>\<otimes> y)"
base: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "base \<in> {..<n} \<rightarrow> {..<p}"
Min_false1: fixes w :: "nat"   and rf :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "\<not> Min {uu. uu \<le> w \<and> 0 < rec_exec rf (xs @ [uu])} \<le> w"   and "x \<le> w"   and "0 < rec_exec rf (xs @ [x])" shows "False"
simps_4: fixes x :: "('a, 'b) Node"   and xs :: "('a, 'b) Node list"   and v :: "('a, 'b) Node" shows "FingerTreeStruc.app3 (Single x) xs (Single v) = FingerTreeStruc.nlcons x (FingerTreeStruc.lconsNlist xs (Single v))"
assms_2: shows "integral\<^sup>N M f \<noteq> \<infinity>"
basic_trans_rules_24: fixes x :: "'h"   and y :: "'h" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
has_contour_integral_reversepath: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and i :: "complex" assumes "valid_path g"   and "(f has_contour_integral i) g" shows "(f has_contour_integral - i) (reversepath g)"
intro: fixes A :: "('h, 'i) graph_rec_scheme"   and B :: "('j, 'k) graph_rec_scheme"   and R :: "('h \<times> 'j) set" assumes "graph A"   and "graph B"   and "simulation R A B"   and "simulation (R\<inverse>) B A" shows "bisimulation R A B"
fun_rel_mono: fixes R1 :: "('a \<times> 'b) set"   and R2 :: "('a \<times> 'b) set"   and R3 :: "('c \<times> 'd) set"   and R4 :: "('c \<times> 'd) set" assumes "R1 \<subseteq> R2"   and "R3 \<subseteq> R4" shows "R2 \<rightarrow> R3 \<subseteq> R1 \<rightarrow> R4"
assms_3: fixes \<sigma> :: "'\<sigma>" assumes "I S0 \<sigma>" shows "P \<sigma>"
E_validD_2: fixes v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "(v, e, v') \<in> G.E" shows "v' \<in> G.V"
anno_type_sec_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and upd :: "'Var ModeUpd" shows "anno_type_sec \<Gamma> \<S> P upd \<equiv> case upd of var +=\<^sub>m Mode \<Rightarrow> True | v -=\<^sub>m AsmNoReadOrWrite \<Rightarrow> v \<in> add_anno_dom \<Gamma> \<S> upd \<longrightarrow> the (\<Gamma> v) \<le>:\<^sub>P dma_type v | v -=\<^sub>m _ \<Rightarrow> True"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (trg a)"
quasi_inverse_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 f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f" shows "partial_magma.ide V (bicategory.some_quasi_inverse V H \<a> \<i> src trg f)"
poly_root_factor_3: fixes r :: "'a"   and k :: "'a" shows "(poly [:r, 1::'a:] k = (0::'a)) = (k = - r)"
simps_9: fixes f :: "'e \<Rightarrow> 'd"   and x21 :: "'e"   and x22 :: "'e list" shows "map f (x21 # x22) = f x21 # map f x22"
add_diff_inverse: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "a + (b - a) = b"
LD'_def: fixes r :: "('b \<times> 'b) set"   and \<beta> :: "'b"   and \<alpha> :: "'b"   and \<sigma>1 :: "'b list"   and \<sigma>2 :: "'b list"   and \<sigma>3 :: "'b list"   and \<tau>1 :: "'b list"   and \<tau>2 :: "'b list"   and \<tau>3 :: "'b list" shows "LD' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<tau>1 \<tau>2 \<tau>3 = (LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LD_1' r \<alpha> \<beta> \<tau>1 \<tau>2 \<tau>3)"
n_c_narrow: fixes C1 :: "ivl st option acom"   and C2 :: "ivl st option acom" assumes "strip C1 = strip C2"   and "top_on\<^sub>c C1 (- vars C1)"   and "top_on\<^sub>c C2 (- vars C2)"   and "C2 \<le> C1"   and "C1 \<triangle> C2 < C1" shows "n_c (C1 \<triangle> C2) < n_c C1"
supto_is_snil_conv_2: fixes i :: "int"   and j :: "int" shows "([::] = supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j)) = (j < i)"
min_maxsimpchainD_maxsimpchain: fixes xs :: "'a set list" assumes "min_gallery xs" shows "folding.gallery xs"
simps: fixes a :: "'b llist"   and A :: "'b set" shows "(a \<in> A\<^sup>\<infinity>) = (a = LNil \<or> (\<exists>l aa. a = aa ## l \<and> l \<in> A\<^sup>\<infinity> \<and> aa \<in> A))"
sas_plus_problem_to_strips_problem_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem" shows "\<phi> \<Psi>  \<equiv> let vs = concat (map (\<lambda>v. map (\<lambda>as. as) (possible_assignments_for \<Psi> v)) (\<Psi>\<^sub>\<V>\<^sub>+)); ops = map (sasp_op_to_strips \<Psi>) (\<Psi>\<^sub>\<O>\<^sub>+); I = \<phi>\<^sub>S \<Psi> \<Psi>\<^sub>I\<^sub>+ in Let (\<phi>\<^sub>S \<Psi> \<Psi>\<^sub>G\<^sub>+) (problem_for vs ops I)"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b H"   and y :: "'d H"   and Ra :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_H R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_H x; yb \<in> set_H y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_H Ra x y"
minimal_antichain_path_weightp_member: fixes s :: "'lbl"   and l1 :: "'vtx"   and l2 :: "'vtx" assumes "s \<in> {xs. path_weightp l1 l2 xs}" shows "\<exists>u. u \<in> minimal_antichain {xs. path_weightp l1 l2 xs} \<and> u \<le> s"
block_size_gt_t: fixes k :: "int" assumes "k \<in> \<K>" shows "2 \<le> k"
finsum_vec_closed: fixes f :: "'a \<Rightarrow> 'b vec"   and A :: "'a set"   and n :: "nat" assumes "f \<in> A \<rightarrow> carrier_vec n" shows "finsum_vec TYPE('b) n f A \<in> carrier_vec n"
mcont_applyI: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and F :: "'f \<Rightarrow> 'g"   and x :: "'h" assumes "mcont luba orda lubb ordb F" shows "mcont (fun_lub luba) (fun_ord orda) lubb ordb (\<lambda>f. F (f x))"
vfsequence_vcons_vdomain: fixes x :: "V" shows "\<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs)"
assms_1: shows "initmissing (netgmap sr s) \<in> oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"
that: fixes p :: "('a \<times> 'a) list" assumes "path g u p v" shows "thesis"
dagger_unfoldl_eq: fixes x :: "'b \<times> 'c" shows "sup_prod one_prod (times_prod' x (prod.qstar x)) = prod.qstar x"
not_one_le_zero_qdelta: shows "\<not> 1 \<le> 0"
fps_lr_inverse_mult_ring1_1: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a"   and g :: "'a fps" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * x = (1::'a)"   and "y * fps_nth g 0 = (1::'a)"   and "fps_nth g 0 * y = (1::'a)" shows "fps_left_inverse (f * g) (y * x) = fps_left_inverse g y * fps_left_inverse f x"
1: fixes u :: "real"   and A :: "real \<Rightarrow> bool"   and v :: "real" assumes "\<forall>n\<ge>u. A n"   and "\<forall>n\<le>v. A n" shows "\<exists>b. \<forall>x. b \<le> \<bar>x\<bar> \<longrightarrow> A x"
os_pop_rule: fixes xs :: "'a list"   and r :: "'a node ref option" assumes "xs \<noteq> []" shows "<os_list xs r> os_pop r <\<lambda>(x, r'). os_list (tl xs) r' * the r \<mapsto>\<^sub>r Node x r' * \<up> (x = hd xs)>"
induct: fixes x1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and x2 :: "'d"   and x3 :: "('c \<Rightarrow> real) set"   and x4 :: "'d"   and x5 :: "('c \<Rightarrow> real) set"   and P :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> ('c \<Rightarrow> real) set \<Rightarrow> 'd \<Rightarrow> ('c \<Rightarrow> real) set \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>x4, x5\<rangle>"   and "\<And>A l Z. P A l Z l Z"   and "\<And>A l Z l' Z' l'' Z''. \<lbrakk>A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>; P A l Z l' Z'; A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>l'', Z''\<rangle>\<rbrakk> \<Longrightarrow> P A l Z l'' Z''" shows "P x1 x2 x3 x4 x5"
simps: fixes q :: "'b"   and n :: "nat" shows "senior_states q n = {p. \<exists>x y. oldest_token p n = Some y \<and> oldest_token q n = Some x \<and> y < x \<and> \<not> sink p}"
A: shows "A \<in> sets (Pi\<^sub>M I M)"
sup2: fixes B :: "'e"   and k :: "'e"   and b :: "'e"   and a :: "'e" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
assms: shows "diagram J (\<cdot>\<^sub>A) D"
measurable_C: fixes n :: "nat"   and m :: "nat" shows "C n m \<in> Pi\<^sub>M {0..<n} (\<lambda>_. M) \<rightarrow>\<^sub>M subprob_algebra (Pi\<^sub>M {0..<n + m} (\<lambda>_. M))"
list_all2_update_cong: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list"   and x :: "'e"   and y :: "'f"   and i :: "nat" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs[i := x]) (ys[i := y])"
assms: fixes v :: "nat" assumes "v \<in> V"   and "(v, u) \<notin> E" shows "g (v, u) = (0::'b)"
point_linorder_matrix_unique: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.point f = (\<exists>!i. \<forall>j. f (i, j) = top \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
is_weak_pmap_def: fixes f :: "'e \<Rightarrow> 'f"   and C :: "('g set \<times> 'g set \<times> 'g set) \<times> 'e set \<times> ('e \<times> 'g \<times> 'e) set"   and A :: "('g set \<times> 'g set \<times> 'g set) \<times> 'f set \<times> ('f \<times> 'g \<times> 'f) set" shows "is_weak_pmap f C A \<equiv> (\<forall>s\<in>starts_of C. f s \<in> starts_of A) \<and> (\<forall>s t a. reachable C s \<and> (s, a, t) \<in> trans_of C \<longrightarrow> (if a \<in> externals (asig_of C) then (f s, a, f t) \<in> trans_of A else f s = f t))"
subgame_VV_subset: fixes p :: "Player"   and V' :: "'a set" shows "(if p = Even then V0\<^bsub>subgame V'\<^esub> else V\<^bsub>subgame V'\<^esub> - V0\<^bsub>subgame V'\<^esub>) \<subseteq> VV p"
pderiv_minus: fixes p :: "'b poly" shows "pderiv (- p) = - pderiv p"
rev: fixes xs :: "'b list" shows "sum_list (rev xs) = sum_list xs"
nvalid_augment_Faults: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and n :: "nat"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and F' :: "'g set" assumes "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F'\<^esub> P c Q,A"
inverse_mod_coprime_exp: fixes p :: "int"   and n :: "nat"   and x :: "int" assumes "p ^ n = p ^ n"   and "prime p"   and "n \<noteq> 0"   and "comm_monoid_mult_class.coprime x p" shows "M (inverse_mod x (p ^ n) * x) = 1"
assms: shows "valid_pratt_tree (Pratt_Node (n, a, ts)) \<equiv> True"
list_of_dlist_inverse: fixes x :: "'a dlist" shows "Abs_dlist (list_of_dlist x) = x"
arg_min_on_def: fixes less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and S :: "'d set" shows "ord.arg_min_on less f S = ord.arg_min less f (\<lambda>x. x \<in> S)"
inject_1: fixes x11 :: "vrs"   and x12 :: "ty"   and y11 :: "vrs"   and y12 :: "ty" shows "(VarB_Rep x11 x12 = VarB_Rep y11 y12) = (x11 = y11 \<and> x12 = y12)"
assms: shows "valid (\<Delta> @ [(x, T)] @ \<Gamma>)"
IH: fixes i :: "nat" shows "\<exists>J\<subseteq>I. \<exists>X\<in>sets (Pi\<^sub>M J M). countable J \<and> K i = prod_emb I M J X"
nth_rev_upt: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j + k < i" shows "[i>..j] ! k = i - 1 - k"
tauLaw3StatImpRight: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', \<alpha>\<cdot>P, \<alpha>\<cdot>(\<tau>.Q)) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>P \<lessapprox><Rel> \<alpha>\<cdot>(\<tau>.Q)"
dist_norm: fixes x :: "'b"   and y :: "'b" shows "dist x y = norm (x - y)"