hcju/isabelleps huggingface dataset:
fact_numeral: fixes k :: "num" shows "fact (numeral k) = numeral k * fact (pred_numeral k)"
sel_1: fixes xs :: "('f, 'g) tllist" assumes "\<not> is_TNil xs" shows "lhd (llist_of_tllist xs) = (case xs of TCons x xs' \<Rightarrow> x)"
isCont_norm: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "isCont f a" shows "isCont (\<lambda>x. norm (f x)) a"
has_integral_reflect_lemma_real: fixes f :: "real \<Rightarrow> 'b"   and i :: "'b"   and a :: "real"   and b :: "real" assumes "(f has_integral i) {a..b}" shows "((\<lambda>x. f (- x)) has_integral i) {- b..- a}"
sorted_lower: shows "sorted_sublist_map R h xs'' lo (p - Suc 0)"
abs_eq: fixes xa :: "'a"   and x :: "'a poly" shows "smult xa x = Abs_poly (\<lambda>n. xa * coeff x n)"
lookup_default_def: fixes d :: "'c"   and m :: "('d, 'c) mapping"   and k :: "'d" shows "Mapping.lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
Rcd_type2: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and T :: "POPLmarkRecord.type"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> Rcd fs : T"   and "\<Gamma> \<turnstile> T <: RcdT fTs" shows "\<Gamma> \<turnstile> map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs [:] fTs"
DSTEP_star_comp2: fixes l :: "('c + 'd) list"   and l' :: "('c + 'd) list"   and G :: "('c \<times> ('c, 'd) RHS) list"   and r :: "('c + 'd) list"   and r' :: "('c + 'd) list" assumes "l -G\<rightarrow>\<^sup>* l'"   and "r -G\<rightarrow>\<^sup>* r'" shows "l \<cdot> r -G\<rightarrow>\<^sup>* l' \<cdot> r'"
lower_split_interval1: fixes X :: "'b interval"   and m :: "'b" shows "lower (fst (split_interval X m)) = min (lower X) m"
remdups_gen_subset: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "set (Missing_List.remdups_gen f xs) \<subseteq> set xs"
res: assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; True\<rbrakk> \<Longrightarrow> True" shows "set_iterator (set_iterator_image g it) S'"
eq'_2: fixes p :: "nat" assumes "prime p" shows "multiplicity p (\<Prod>A1) = multiplicity p (\<Prod>A2)"
comp_null_2: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.comp f S.null = S.null"
WT_gpv_bindI: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('g, 'd, 'e) gpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "\<And>x. x \<in> results_gpv \<Gamma> gpv \<Longrightarrow> \<Gamma> \<turnstile>g f x \<surd>" shows "\<Gamma> \<turnstile>g gpv \<bind> f \<surd>"
validity2: fixes n :: "nat"   and n' :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> n'"   and "\<exists>i\<ge>n'. active c (t i)"   and "\<forall>k\<ge>n. k < n' \<longrightarrow> \<not> active c (t k)"   and "dynamic_component.eval tCMP active c t t' n' \<gamma>" shows "dynamic_component.eval tCMP active c t t' n \<gamma>"
finalize_satisfies: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i" assumes "local.wf idx \<phi>"   and "#\<^sub>V \<AA> = idx"   and "lformula \<phi>" shows "(\<AA> \<Turnstile>\<^sub>b finalize idx \<phi>) = (\<AA> \<Turnstile> \<phi>)"
exists_smooth_on: fixes x :: "'a" assumes "x \<in> src.carrier" shows "\<exists>c1\<in>src.atlas. \<exists>c2\<in>dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
termi_merge_c1: shows "\<Gamma>\<turnstile>merge_guards c1 \<down> Normal s"
copm_2: shows "comm_monoid_mult_class.coprime (lead_coeff g) m"
m_defsAll_5: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC theta c d s t I0 P F \<equiv> mC_ZOC_part c d s t I0 P F \<and> inj_on F P \<and> mC_ZOC_wt c d s t I0 P F \<and> mC_ZOC_eff_cont0 theta c d s t I0 F \<and> mC_ZOC_eff_cont theta c d s t I0 P F"
simps: fixes n :: "nat"   and l :: "nat" shows "\<Phi> (n, l) = (if f2'' * real l \<le> real n then ai * (real n - f2'' * real l) else if real n \<le> f1'' * real l \<and> int l0 \<le> \<lfloor>real l / c\<rfloor> then ad * (f1'' * real l - real n) else 0)"
in_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and I :: "nat set" shows "(brnL cl n + i \<in> local.shift cl n I) = (i \<in> I)"
ind: fixes b :: "'a"   and c :: "'a" assumes "cbelow_on A ord z rel a b"   and "rel b c"   and "c \<in> A"   and "ord c z"   and "P b" shows "P c"
iEx_cong: fixes I :: "nat set"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "\<box> t I. f t = g t" shows "(\<diamond> t I. P (f t) t) = (\<diamond> t I. P (g t) t)"
seqllI: fixes P :: "('e \<times> 'f) \<times> 'g \<times> 'e \<times> 'f \<Rightarrow> bool"   and \<sigma> :: "'h \<Rightarrow> 'e"   and i :: "'h"   and p :: "'f"   and a :: "'g"   and \<sigma>' :: "'h \<Rightarrow> 'e"   and p' :: "'f" assumes "P ((\<sigma> i, p), a, \<sigma>' i, p')" shows "seqll i P ((\<sigma>, p), a, \<sigma>', p')"
dtle: fixes a :: "nat"   and aa :: "nat" shows "a div aa * aa \<le> a"
ConjS: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>& \<psi> in v] = ([\<phi> in v] \<and> [\<psi> in v])"
simps_2: fixes \<mu> :: "'a" shows "E.Src \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>src\<^sub>B \<mu>\<^bold>\<rangle>\<^sub>0"
domain_eq: shows "Domainp pcr_moebius_mat = (\<lambda>M. mat_det M \<noteq> 0)"
map_iterator_rev_linord_rule_P: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'a \<Rightarrow> 'c option"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<times> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_rev_linord less_eq it (map_to_set m)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<And>k'. k' \<in> it \<Longrightarrow> less_eq k' k; \<And>k'. k' \<in> dom m - it \<Longrightarrow> less_eq k k'\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<And>k k'. \<lbrakk>k \<in> dom m - it; k' \<in> it\<rbrakk> \<Longrightarrow> less_eq k' k\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
maximal_existence_flow_2: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and x :: "real \<Rightarrow> 'a"   and K :: "real set"   and t0 :: "real"   and x0 :: "'a"   and t :: "real" assumes "ll_on_open_it T f X"   and "(x solves_ode f) K X"   and "x t0 = x0"   and "is_interval K"   and "t0 \<in> K"   and "K \<subseteq> T"   and "t \<in> K" shows "ll_on_open.flow T f X t0 x0 t = x t"
trancl_insert: fixes y :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set" shows "(insert (y, x) r)\<^sup>+ = r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
zero_le_iff_nprt_id: 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq a zero = (lattice_ab_group_add.nprt zero inf a = a)"
reaction_mono: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and vl :: "'value list"   and s1 :: "'state"   and vl1 :: "'value list"   and \<Delta>' :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "reaction \<Delta> s vl s1 vl1"   and "\<And>s vl s1 vl1. \<Delta> s vl s1 vl1 \<Longrightarrow> \<Delta>' s vl s1 vl1" shows "reaction \<Delta>' s vl s1 vl1"
induct: fixes x1 :: "int"   and x2 :: "(int \<times> Instr) list"   and x3 :: "Prim"   and x4 :: "(int \<times> Instr) list \<times> int"   and P :: "int \<Rightarrow> (int \<times> Instr) list \<Rightarrow> Prim \<Rightarrow> (int \<times> Instr) list \<times> int \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> compilePrim"   and "\<And>l code i. P l code (IntPrim i) (code[l\<mapsto>const (IVal i)], l + 1)"   and "\<And>l code f x. P l code (UnPrim f x) (code[l\<mapsto>load x][l + 1\<mapsto>unop f], l + 2)"   and "\<And>l code f x y. P l code (BinPrim f x y) (code[l\<mapsto>load x][l + 1\<mapsto>load y][l + 2\<mapsto>binop f], l + 3)"   and "\<And>l code. P l code NilPrim (code[l\<mapsto>const (RVal Nullref)], l + 1)"   and "\<And>l code x y. P l code (ConsPrim x y) (code[l\<mapsto>load y][l + 1\<mapsto>load x][l + 2\<mapsto>new LIST][l + 3\<mapsto>store tmp][l + 4\<mapsto>load tmp][l + 5\<mapsto>putfield LIST HD][l + 6\<mapsto>load tmp][l + 7\<mapsto>putfield LIST TL][l + 8\<mapsto>load tmp], l + 9)"   and "\<And>l code f. P l code (CallPrim f []) (code[l\<mapsto>invokeS (fst f) (snd f)], l + 1)"   and "\<And>l code x f args OUT. \<lbrakk>(l + 1, code[l\<mapsto>load x], CallPrim f args, OUT) \<in> compilePrim; P (l + 1) code[l\<mapsto>load x] (CallPrim f args) OUT\<rbrakk> \<Longrightarrow> P l code (CallPrim f (x # args)) OUT" shows "P x1 x2 x3 x4"
set_Macaulay_red: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "set (punit.Macaulay_red (punit.Keys_to_list fs) fs) = set (punit.Macaulay_list fs) - {p. punit.lt p \<in> punit.lt_set (set fs)}"
SModule_restrict_scalars: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and S :: "'a set" assumes "RModule R smult M"   and "Ring1 S \<and> S \<subseteq> R" shows "RModule S smult M"
image_cong: fixes M :: "'h set"   and N :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
Vfrom_sup: fixes a :: "V"   and i :: "V"   and j :: "V" shows "Vfrom a (i \<squnion> j) = Vfrom a i \<squnion> Vfrom a j"
notStuck_SeqD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<and> (\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow>\<notin>{Stuck})"
assms_7: assumes "m < n \<and> x < y"   and "x \<noteq> m"   and "x \<noteq> n"   and "y = m" shows "P (arr x) (arr n)"
big_step_simps_1: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (SKIP, s) \<Rightarrow> t = (t = s)"
VectorSpaceEnd_axioms_def: fixes V :: "'a set"   and T :: "'a \<Rightarrow> 'a" shows "VectorSpaceEnd_axioms V T \<equiv> T ` V \<subseteq> V"
add_comm: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = ell_field_class.add a p\<^sub>2 p\<^sub>1"
subset_dual1: fixes X :: "'b set"   and Y :: "'b set" shows "(X \<subseteq> Y) = (\<partial> ` X \<subseteq> \<partial> ` Y)"
comp_ide_self: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "C a a = a"
abs_eq: fixes x :: "'a \<Rightarrow> real" shows "sq_mtx_diag x = to_mtx (diag_mat x)"
subdigraphI: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "\<And>a. a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr> \<Longrightarrow> a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<And>a b f. f : a \<mapsto>\<^bsub>\<BB>\<^esub> b \<Longrightarrow> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
tranclp_into_rtranclp: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
local_flow_axioms_def: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "local_flow_axioms f T S \<phi> \<equiv> (\<forall>t s. t \<in> T \<longrightarrow> s \<in> S \<longrightarrow>  D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--t}) \<and> (\<forall>s. s \<in> S \<longrightarrow> \<phi> 0 s = s) \<and> (\<forall>t s. t \<in> T \<longrightarrow> s \<in> S \<longrightarrow> (\<lambda>t. \<phi> t s) \<in> {0--t} \<rightarrow> S)"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<le> a \<squnion> b"
lim_real_of_ereal: fixes f :: "'b \<Rightarrow> ereal"   and x :: "real"   and net :: "'b filter" assumes "(f \<longlongrightarrow> ereal x) net" shows "((\<lambda>x. real_of_ereal (f x)) \<longlongrightarrow> x) net"
g0_2_f0: shows "cmod (g 0) \<le> 2 + cmod (f 0)"
ldrop_lmap: fixes n :: "enat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "ldrop n (lmap f xs) = lmap f (ldrop n xs)"
crdt_ops_rem_last: fixes xs :: "('c \<times> 'd) list"   and x :: "'c \<times> 'd"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops (xs @ [x]) deps" shows "crdt_ops xs deps"
infsetsum_reindex_bij_betw: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "bij_betw g A B" shows "(\<Sum>\<^sub>ax\<in>A. f (g x)) = infsetsum f B"
coherent_Lunit_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "coherent \<^bold>\<l>\<^bold>[t\<^bold>]"
a_set_child_nodes_tups_def: shows "CD.a_set_child_nodes_tups \<equiv> [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
Func_option_def: fixes A :: "'a set"   and B :: "'b set" shows "Func_option A B \<equiv> {f. (\<forall>a. (f a \<noteq> None) = (a \<in> A)) \<and> (\<forall>a\<in>A. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
add_block_fin_cond: fixes b :: "'a set" assumes "b \<subseteq> bl" shows "finite_incidence_system bl (derived_incidence_sys.add_block b)"
fps_inverse_zero': assumes "inverse (0::'a) = (0::'a)" shows "inverse 0 = 0"
af_subsequence_R: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "af (\<phi> R\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) and\<^sub>n af \<phi> (prefix (Suc n) w) or\<^sub>n af (\<phi> R\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n])"
not_None_eq: fixes x :: "'f option" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
simplicial_chain_cmul: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and a :: "int" assumes "simplicial_chain p S c" shows "simplicial_chain p S (frag_cmul a c)"
mset_msort: fixes xs :: "'a list" shows "mset (msort xs) = mset xs"
elims: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "ins x = y" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "y = (n + 1, if n < l then l else if l = 0 then 1 else nat \<lceil>c * real l\<rceil>)"
consistentE: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "consistent (ae, a, as) (\<Gamma>, e, S)" obtains "a_consistent (ae, a, as) (\<Gamma>, e, S)" and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0"
action_abst_repcs: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "abst (action (repcs s cfg)) = action cfg"
order_def: fixes a :: "'a"   and p :: "'a poly" shows "order a p = (LEAST n. \<not> [:- a, 1::'a:] ^ Suc n dvd p)"
pathVertices_edge: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and e :: "nat \<times> nat" assumes "isPath s p t"   and "e \<in> set p" shows "\<exists>vs1 vs2. pathVertices_fwd s p = vs1 @ fst e # snd e # vs2"
ge_const_equiv: shows "(\<sim>\<^sub>C) \<le> (\<sim>\<^sub>P)"
bisimCoinductAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi> \<and> \<Psi> \<rhd> P \<leadsto>[(X \<union> bisim)] Q \<and> (\<forall>\<Psi>'. (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<or> \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q) \<and> ((\<Psi>, Q, P) \<in> X \<or> \<Psi> \<rhd> Q \<sim> P)" shows "\<Psi> \<rhd> P \<sim> Q"
mult_of_int_commute: fixes x :: "int"   and y :: "'a" shows "of_int x * y = y * of_int x"
scaleR1: fixes e :: "real"   and xr :: "real \<times> real"   and P :: "real \<times> real" assumes "0 < e"   and "ccw' 0 xr P" shows "ccw' 0 (e *\<^sub>R xr) P"
ltakeWhile_lappend2: fixes xs :: "'b llist"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b llist" assumes "lset xs \<subseteq> {x. P x}" shows "ltakeWhile P (lappend xs ys) = lappend xs (ltakeWhile P ys)"
\<phi>: shows "\<forall>(\<phi>, n, s)\<in>set xs. numbound0 s \<and> isrlfm \<phi>"
hseq_char: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "S.hseq \<nu> \<mu> = (S.arr \<mu> \<and> S.arr \<nu> \<and> S.src \<nu> = S.trg \<mu>)"
Macaulay_mat_map_key_PP: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.punit.Macaulay_mat (map (Poly_Mapping.map_key PP) fs) = punit.Macaulay_mat fs"
n_entails_bot: shows "\<G>_Fset N \<Turnstile>G \<G>_Fset {B}"
mprod_memTr: fixes n :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. e j \<in> Zset) \<and> (\<forall>j\<le>n. f j \<in> carrier K - {\<zero>}) \<longrightarrow> mprod_exp K e f n \<in> carrier K - {\<zero>}"
map_cong: fixes x :: "'c tree"   and ya :: "'c tree"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> set_tree ya \<Longrightarrow> f z = g z" shows "map_tree f x = map_tree g ya"
input_stored_in_reg_is_generalisation: fixes t' :: "transition"   and t :: "transition"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and i :: "nat"   and r :: "nat" assumes "input_stored_in_reg t' t e = Some (i, r)" shows "is_generalisation_of t' t i r"
rel_mono: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_bset R \<le> rel_bset Ra"
sWlsAbsNE_def: fixes SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme" shows "sWlsAbsNE SEM \<equiv> \<forall>us s. isInBar (us, s) \<longrightarrow> (\<exists>sA. sWlsAbs SEM (us, s) sA)"
cubic_root_radical_sqrt: fixes a :: "real"   and b :: "real"   and c :: "real"   and e :: "expr"   and n :: "nat" assumes "a \<in> \<rat>"   and "b \<in> \<rat>"   and "c \<in> \<rat>"   and "card (radicals e) = n"   and "\<lbrace>e\<rbrace> ^ 3 + a * \<lbrace>e\<rbrace>\<^sup>2 + b * \<lbrace>e\<rbrace> + c = 0" shows "\<exists>x\<in>\<rat>. x ^ 3 + a * x\<^sup>2 + b * x + c = 0"
extend_eval: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e'' :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and b'' :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e'',s'',b''\<rangle>"   and "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_7: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) \<union> set X"
swapidseq_id_iff: fixes p :: "'a \<Rightarrow> 'a" shows "swapidseq 0 p = (p = id)"
Brouwer_degree2_nullhomotopic: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> real" assumes "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f (\<lambda>x. a)" shows "Brouwer_degree2 p f = 0"
takeWhile_replicate_append: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and x :: "nat"   and ys :: "'a list" assumes "P a" shows "takeWhile P (a \<up> x @ ys) = a \<up> x @ takeWhile P ys"
DE_sys_valid: fixes ODE :: "('sf, 'sz) ODE" assumes "{Inl vid1, Inr vid1} \<inter> BVO ODE = {}" shows "valid ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]]P pid1 \<leftrightarrow> ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]][[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))"
assms_3: fixes a :: "'a" assumes "A.ide a" shows "\<sigma> a = \<sigma>' a"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'b" assumes "finite (Graph.V c)" shows "Finite_Graph c"