theory_file stringlengths 3 95 | lemma_name stringlengths 5 250 | lemma_command stringlengths 15 21.3k ⌀ | lemma_object stringlengths 5 62.8k | template stringlengths 7 63.6k | output_key stringclasses 1
value | input stringlengths 21 121k | output stringlengths 21 21.3k |
|---|---|---|---|---|---|---|---|
Analysis/Complete_Measure | Complete_Measure.completion_ex_borel_measurable_real | lemma completion_ex_borel_measurable_real:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable (completion M)"
shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)" | ?g \<in> borel_measurable (completion ?M) \<Longrightarrow> \<exists>g'\<in>borel_measurable ?M. AE x in ?M. ?g x = g' x | x_1 \<in> ?H1 (?H2 x_2) \<Longrightarrow> \<exists>y_0\<in>?H1 x_2. ?H3 x_2 (\<lambda>y_1. x_1 y_1 = y_0 y_1) | lemma_command | ###symbols
Complete_Measure.locally_determined_measure
Complete_Measure.completion
Measure_Space.distr
Measure_Space.almost_everywhere
Borel_Space.borel_measurable
Complete_Measure.main_part
###defs
abbreviation almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
"almost_e... | ###output
lemma completion_ex_borel_measurable_real:
fixes g :: "'a \<Rightarrow> real"
assumes g: "g \<in> borel_measurable (completion M)"
shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
###end |
Nonstandard_Analysis/Star | StarDef.star_n_eq_iff | null | (star_n ?X = star_n ?Y) = (\<forall>\<^sub>F n in \<U>. ?X n = ?Y n) | (?H1 x_1 = ?H1 x_2) = ?H2 (\<lambda>y_0. x_1 y_0 = x_2 y_0) ?H3 | lemma_command | ###symbols
Relation.Image
StarDef.FreeUltrafilterNat
Filter.eventually
StarDef.star_n
###defs
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90)
where "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
definition FreeUltrafilterNat :: "nat filter" (\<open>\<U>\<close... | ###output
None
###end |
Nonstandard_Analysis/StarDef | StarDef.star_of_compare_numeral(9) | null | (star_of (- numeral ?k) = star_of ?y) = (- numeral ?k = ?y) | (?H1 (?H2 (?H3 x_1)) = ?H1 x_2) = (?H2 (?H3 x_1) = x_2) | lemma_command | ###symbols
StarDef.star_n
Groups.uminus_class.uminus
StarDef.Standard
StarDef.star_of
Finite_Set.finite
Num.numeral_class.numeral
StarDef.star
###defs
definition star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star"
where "star_n X = Abs_star (starrel `` {X})"
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a"... | ###output
None
###end |
Library/Multiset | Multiset.multi_nonempty_split | lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A" | ?M \<noteq> {#} \<Longrightarrow> \<exists>A a. ?M = add_mset a A | x_1 \<noteq> ?H1 \<Longrightarrow> \<exists>y_0 y_1. x_1 = ?H2 y_1 y_0 | lemma_command | ###symbols
Multiset.empty_mset
Multiset.add_mset
Cancellation.iterate_add
List.list_update
###defs
abbreviation empty_mset :: \<open>'a multiset\<close> (\<open>{#}\<close>)
where \<open>empty_mset \<equiv> 0\<close>
definition iterate_add :: \<open>nat \<Rightarrow> 'a::cancel_comm_monoid_add \<Rightarrow> 'a\<close... | ###output
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A"
###end |
Core_SC_DOM/common/monads/NodeMonad | NodeMonad.node_ptr_kinds_small | lemma node_ptr_kinds_small:
assumes "\<And>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"
shows "node_ptr_kinds h = node_ptr_kinds h'" | (\<And>object_ptr. preserved (get_M object_ptr RObject.nothing) ?h ?h') \<Longrightarrow> node_ptr_kinds ?h = node_ptr_kinds ?h' | (\<And>y_0. ?H1 (?H2 y_0 ?H3) x_1 x_2) \<Longrightarrow> ?H4 x_1 = ?H4 x_2 | lemma_command | ###symbols
Complete_Lattices.Union
ObjectMonad.get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t
Heap_Error_Monad.returns_result
ObjectClass.RObject.nothing
Heap_Error_Monad.preserved
NodeMonad.get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e
NodeClass.node_ptr_kinds
###defs
abbreviation Union :: "'a set set \<Rightarrow> 'a ... | ###output
lemma node_ptr_kinds_small:
assumes "\<And>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"
shows "node_ptr_kinds h = node_ptr_kinds h'"
###end |
List-Infinite/CommonSet/SetIntervalCut | SetIntervalCut.cut_ge_subset_mono | null | ?A \<subseteq> ?B \<Longrightarrow> ?A \<down>\<ge> ?t \<subseteq> ?B \<down>\<ge> ?t | ?H1 x_1 x_2 \<Longrightarrow> ?H1 (?H2 x_1 x_3) (?H2 x_2 x_3) | lemma_command | ###symbols
Set.union
Set.subset_eq
SetIntervalCut.cut_ge
###defs
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
| ###output
None
###end |
AODV/variants/d_fwdrreqs/D_Seq_Invariants | D_Seq_Invariants.hop_count_positive | lemma hop_count_positive:
"paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, _). \<forall>ip\<in>kD (rt \<xi>). the (dhops (rt \<xi>) ip) \<ge> 1)" | paodv ?i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, uu_). \<forall>ip\<in>kD (rt \<xi>). 1 \<le> the (dhops (rt \<xi>) ip)) | ?H1 (?H2 x_1) (?H3 ?H4 (?H5 (\<lambda>y_0 y_1. \<forall>y_2\<in>?H6 (?H7 y_0). ?H8 \<le> ?H9 (?H10 (?H7 y_0) y_2)))) | lemma_command | ###symbols
D_Aodv_Data.nhop
D_Aodv_Data.kD
Groups.one_class.one
Product_Type.prod.case_prod
Option.option.the
D_Aodv.\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V
D_Aodv.state.rt
D_Aodv_Data.dhops
D_Aodv.paodv
Invariants.any_invariant
AWN_Invariants.onl
###defs
class one =
fixes one :: 'a ("1")
definition "prod = {f. \<... | ###output
lemma hop_count_positive:
"paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, _). \<forall>ip\<in>kD (rt \<xi>). the (dhops (rt \<xi>) ip) \<ge> 1)"
###end |
HOL-CSPM/MultiSync | MultiSync.MultiInter_BOT_absorb | null | ?m \<in># ?M \<Longrightarrow> ?P ?m = \<bottom> \<Longrightarrow> MultiInter ?M ?P = \<bottom> | \<lbrakk>?H1 x_1 x_2; x_3 x_1 = ?H2\<rbrakk> \<Longrightarrow> ?H3 x_2 x_3 = ?H2 | lemma_command | ###symbols
Multiset.member_mset
MultiSync.MultiInter
Pcpo.pcpo_class.bottom
###defs
abbreviation member_mset :: \<open>'a \<Rightarrow> 'a multiset \<Rightarrow> bool\<close>
where \<open>member_mset a M \<equiv> a \<in> set_mset M\<close>
abbreviation MultiInter :: \<open>['a multiset, 'a \<Rightarrow> 'b process] \... | ###output
None
###end |
Cook_Levin/Arithmetic | Arithmetic.canonical_tl | lemma canonical_tl: "canonical (x # xs) \<Longrightarrow> canonical xs" | canonical (?x # ?xs) \<Longrightarrow> canonical ?xs | ?H1 (?H2 x_1 x_2) \<Longrightarrow> ?H1 x_2 | lemma_command | ###symbols
Groups.plus_class.plus
List.list.Cons
Arithmetic.canonical
Rings.modulo_class.modulo
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred... | ###output
lemma canonical_tl: "canonical (x # xs) \<Longrightarrow> canonical xs"
###end |
LOFT/Semantics_OpenFlow | Semantics_OpenFlow.no_overlap_not_unefined | lemma no_overlap_not_unefined: "check_no_overlap \<gamma> ft \<Longrightarrow> OF_same_priority_match2 \<gamma> ft p \<noteq> Undefined" | check_no_overlap ?\<gamma> ?ft \<Longrightarrow> OF_same_priority_match2 ?\<gamma> ?ft ?p \<noteq> Undefined | ?H1 x_1 x_2 \<Longrightarrow> ?H2 x_1 x_2 x_3 \<noteq> ?H3 | lemma_command | ###symbols
Semantics_OpenFlow.OF_same_priority_match2
Semantics_OpenFlow.flowtable_behavior.Undefined
Semantics_OpenFlow.check_no_overlap
###defs
definition OF_same_priority_match2 :: "('m, 'p) field_matcher \<Rightarrow> ('m, 'a) flowtable \<Rightarrow> 'p \<Rightarrow> 'a flowtable_behavior" where
"OF_same_priority... | ###output
lemma no_overlap_not_unefined: "check_no_overlap \<gamma> ft \<Longrightarrow> OF_same_priority_match2 \<gamma> ft p \<noteq> Undefined"
###end |
HOL-CSP/CSP_Laws | CSP_Laws.write0_read_non_Sync | lemma write0_read_non_Sync:
"\<lbrakk>d \<in> S; \<And>y. c y \<notin> S\<rbrakk> \<Longrightarrow> (d \<rightarrow> P) \<lbrakk>S\<rbrakk> (c\<^bold>?x \<rightarrow> Q x) = c\<^bold>?x \<rightarrow> ((d \<rightarrow> P) \<lbrakk>S\<rbrakk> Q x)" | ?d \<in> ?S \<Longrightarrow> (\<And>y. ?c y \<notin> ?S) \<Longrightarrow> ?d \<rightarrow> ?P \<lbrakk>?S\<rbrakk> read ?c UNIV ?Q = ?c\<^bold>?x \<rightarrow> (?d \<rightarrow> ?P \<lbrakk>?S\<rbrakk> ?Q x) | \<lbrakk>x_1 \<in> x_2; \<And>y_0. ?H1 (x_3 y_0) x_2\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_1 x_4) x_2 (?H4 x_3 ?H5 x_5) = ?H4 x_3 ?H5 (\<lambda>y_2. ?H2 (?H3 x_1 x_4) x_2 (x_5 y_2)) | lemma_command | ###symbols
Mprefix.read
Skip.SKIP
Set.not_member
Set.UNIV
Sync.Sync
Mprefix.write0
###defs
definition read :: "['a \<Rightarrow> 'b,'a set, 'a \<Rightarrow> 'b process] \<Rightarrow> 'b process"
where "read c A P \<equiv> Mprefix(c ` A) (P o (inv_into A c))"
abbreviation not_member
where "not_member x A \<... | ###output
lemma write0_read_non_Sync:
"\<lbrakk>d \<in> S; \<And>y. c y \<notin> S\<rbrakk> \<Longrightarrow> (d \<rightarrow> P) \<lbrakk>S\<rbrakk> (c\<^bold>?x \<rightarrow> Q x) = c\<^bold>?x \<rightarrow> ((d \<rightarrow> P) \<lbrakk>S\<rbrakk> Q x)"
###end |
Nullstellensatz/Nullstellensatz | Nullstellensatz.ideal_ofD | lemma ideal_ofD: "f \<in> \<I> A \<Longrightarrow> a \<in> A \<Longrightarrow> poly_eval a f = 0" | ?f \<in> \<I> ?A \<Longrightarrow> ?a \<in> ?A \<Longrightarrow> poly_eval ?a ?f = (0::?'b) | \<lbrakk>x_1 \<in> ?H1 x_2; x_3 \<in> x_2\<rbrakk> \<Longrightarrow> ?H2 x_3 x_1 = ?H3 | lemma_command | ###symbols
Groups.times_class.times
Groups.zero_class.zero
MPoly_PM.poly_eval
Nullstellensatz.\<I>
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class zero =
fixes zero :: 'a ("0")
definition poly_eval :: "('x \<Rightarrow> 'a) \<Rightarrow> (('x \<Rightarrow>\<^sub>0... | ###output
lemma ideal_ofD: "f \<in> \<I> A \<Longrightarrow> a \<in> A \<Longrightarrow> poly_eval a f = 0"
###end |
Smith_Normal_Form/Diagonal_To_Smith | Diagonal_To_Smith.diagonal_to_Smith_PQ' | lemma diagonal_to_Smith_PQ':
fixes A::"'a::{bezout_ring}^'cols::{mod_type}^'rows::{mod_type}"
assumes A: "isDiagonal A" and ib: "is_bezout_ext bezout"
assumes PBQ: "(P,S,Q) = diagonal_to_Smith_PQ A bezout"
shows "S = P**A**Q \<and> invertible P \<and> invertible Q \<and> Smith_normal_form S" | isDiagonal ?A \<Longrightarrow> is_bezout_ext ?bezout \<Longrightarrow> (?P, ?S, ?Q) = diagonal_to_Smith_PQ ?A ?bezout \<Longrightarrow> ?S = ?P ** ?A ** ?Q \<and> invertible ?P \<and> invertible ?Q \<and> Smith_normal_form ?S | \<lbrakk>?H1 x_1; ?H2 x_2; (x_3, x_4, x_5) = ?H3 x_1 x_2\<rbrakk> \<Longrightarrow> x_4 = ?H4 (?H5 x_3 x_1) x_5 \<and> ?H6 x_3 \<and> ?H7 x_5 \<and> ?H8 x_4 | lemma_command | ###symbols
Smith_Normal_Form.Smith_normal_form
Set.insert
Groups.zero_class.zero
Finite_Cartesian_Product.matrix_matrix_mult
Groups.uminus_class.uminus
Diagonal_To_Smith.diagonal_to_Smith_PQ
Rings2.bezout_ring_class.is_bezout_ext
Finite_Cartesian_Product.invertible
Smith_Normal_Form.isDiagonal
Miscellaneous.ncols
###de... | ###output
lemma diagonal_to_Smith_PQ':
fixes A::"'a::{bezout_ring}^'cols::{mod_type}^'rows::{mod_type}"
assumes A: "isDiagonal A" and ib: "is_bezout_ext bezout"
assumes PBQ: "(P,S,Q) = diagonal_to_Smith_PQ A bezout"
shows "S = P**A**Q \<and> invertible P \<and> invertible Q \<and> Smith_normal_form S"
###end |
Safe_OCL/OCL_Normalization | OCL_Normalization.ArrowCallNE | null | ?\<Gamma> \<turnstile> Call ?src ArrowCall ?call \<Rrightarrow> ?b \<Longrightarrow> (\<And>src\<^sub>2 \<tau> \<sigma> call\<^sub>2. ?b = Call (OperationCall src\<^sub>2 DotCall (Inl (Inl OclAsSetOp)) []) ArrowCall call\<^sub>2 \<Longrightarrow> ?\<Gamma> \<turnstile> ?sr... | \<lbrakk>?H1 x_1 (?H2 x_2 ?H3 x_3) x_4; \<And>y_0 y_1 y_2 y_3. \<lbrakk>x_4 = ?H2 (?H4 y_0 ?H5 (?H6 (?H7 ?H8)) ?H9) ?H3 y_3; ?H1 x_1 x_2 y_0; ?H10 x_1 y_0 y_1; y_1 \<le> ?H11 ?H12 \<or> y_1 \<le> ?H13 ?H14; ?H10 x_1 (?H4 y_0 ?H5 (?H6 (?H7 ?H8)) ?H9) y_2; ?H15 (x_1, y_2) x_3 y_3\<rbrakk> \<L... | lemma_command | ###symbols
Finite_Map.fmempty
OCL_Types.type.Tuple
OCL_Types.element_type
OCL_Typing.typing
OCL_Normalization.normalize
OCL_Syntax.any_unop.OclAsSetOp
OCL_Types.type.Optional
Sum_Type.Inl
OCL_Normalization.normalize_call
OCL_Syntax.call_kind.DotCall
List.list.Nil
OCL_Syntax.call_kind.ArrowCall
Fun.comp
OCL_Syntax.Opera... | ###output
None
###end |
AODV/variants/e_all_abcd/E_Fresher | E_Fresher.invalidate_rtsf_left | lemma invalidate_rtsf_left [simp]:
"\<And>dests dip rt rt'. dests dip = None \<Longrightarrow> (invalidate rt dests \<sqsubset>\<^bsub>dip\<^esub> rt') = (rt \<sqsubset>\<^bsub>dip\<^esub> rt')" | ?dests ?dip = None \<Longrightarrow> (invalidate ?rt ?dests \<sqsubset>\<^bsub>?dip\<^esub> ?rt') = (?rt \<sqsubset>\<^bsub>?dip\<^esub> ?rt') | x_1 x_2 = ?H1 \<Longrightarrow> ?H2 (?H3 x_3 x_1) x_2 x_4 = ?H2 x_3 x_2 x_4 | lemma_command | ###symbols
E_Fresher.rt_strictly_fresher_syn
Option.option.None
E_Fresher.rt_fresh_as_syn
E_Aodv_Data.invalidate
Map.dom
###defs
datatype 'a option =
None
| Some (the: 'a)
definition
dom :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a set" where
"dom m = {a. m a \<noteq> None}"
| ###output
lemma invalidate_rtsf_left [simp]:
"\<And>dests dip rt rt'. dests dip = None \<Longrightarrow> (invalidate rt dests \<sqsubset>\<^bsub>dip\<^esub> rt') = (rt \<sqsubset>\<^bsub>dip\<^esub> rt')"
###end |
CZH_Elementary_Categories/czh_ecategories/CZH_ECAT_Rel | CZH_ECAT_Rel.cat_Rel_cs_simps(26) | null | arr_Rel ?\<alpha> ?T \<Longrightarrow> \<R>\<^sub>\<circ> (?T\<lparr>ArrVal\<rparr>) = ?A \<Longrightarrow> ?T\<lparr>ArrCod\<rparr> = ?A \<Longrightarrow> v11 (?T\<lparr>ArrVal\<rparr>) \<Longrightarrow> ?A \<in>\<^sub>\<circ> Vset ?\<alpha> \<Longrightarrow> ?T \<circ>\<^sub>R\<^sub>e\<^sub>l ?T\<in... | \<lbrakk>?H1 x_1 x_2; ?H2 (?H3 x_2 ?H4) = x_3; ?H3 x_2 ?H5 = x_3; ?H6 (?H3 x_2 ?H4); ?H7 x_3 (?H8 x_1)\<rbrakk> \<Longrightarrow> ?H9 x_2 (?H10 x_2) = ?H11 x_3 | lemma_command | ###symbols
ZFC_Cardinals.app
CZH_DG_Rel.converse_Rel
CZH_DG_Rel.comp_Rel
CZH_DG_Digraph.Dom
CZH_DG_Rel.arr_Rel
CZH_Sets_BRelations.app_vconverse
CZH_DG_Rel.id_Rel
CZH_DG_Rel.ArrCod
CZH_DG_Rel.ArrVal
CZH_Sets_BRelations.app_vrange
CZH_Sets_BRelations.v11
ZFC_in_HOL.Vset
CZH_Sets_Sets.vmember
###defs
definition app :: "[... | ###output
None
###end |
Word_Lib/More_Word | More_Word.word_le_make_less | lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))" | ?y \<noteq> - 1 \<Longrightarrow> (?x \<le> ?y) = (?x < ?y + 1) | x_1 \<noteq> ?H1 ?H2 \<Longrightarrow> (x_2 \<le> x_1) = (x_2 < ?H3 x_1 ?H2) | lemma_command | ###symbols
Groups.uminus_class.uminus
Groups.one_class.one
Groups.plus_class.plus
Word.is_down
###defs
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
class one =
fixes one :: 'a ("1")
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
| ###output
lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))"
###end |
Computational_Algebra/Formal_Power_Series | Formal_Power_Series.fps_cutoff_zero | lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0" | fps_cutoff ?n 0 = 0 | ?H1 x_1 ?H2 = ?H2 | lemma_command | ###symbols
Formal_Power_Series.fps_cutoff
Groups.zero_class.zero
###defs
definition "fps_cutoff n f = Abs_fps (\<lambda>i. if i < n then f$i else 0)"
class zero =
fixes zero :: 'a ("0")
| ###output
lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0"
###end |
Jinja/J/BigStep | BigStep.eval_finalsId | lemma eval_finalsId:
assumes finals: "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>" | finals ?es \<Longrightarrow> ?P \<turnstile> \<langle>?es,?s\<rangle> [\<Rightarrow>] \<langle>?es,?s\<rangle> | ?H1 x_1 \<Longrightarrow> ?H2 x_2 x_1 x_3 x_1 x_3 | lemma_command | ###symbols
Expr.THROW
BigStep.evals
Objects.init_fields
BigStep.finals
List.list.Cons
###defs
definition finals:: "expr list \<Rightarrow> bool" where
"finals es \<equiv> (\<exists>vs. es = map Val vs) \<or> (\<exists>vs r es'. es = map Val vs @ Throw r # es')"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd... | ###output
lemma eval_finalsId:
assumes finals: "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
###end |
Containers/AssocList | AssocList.set_delete | lemma set_delete: "set (delete k al) = set al - {k} \<times> UNIV" | AssocList.set (AssocList.delete ?k ?al) = AssocList.set ?al - {?k} \<times> UNIV | ?H1 (?H2 x_1 x_2) = ?H3 (?H1 x_2) (?H4 (?H5 x_1 ?H6) ?H7) | lemma_command | ###symbols
Set.empty
Set.UNIV
Set.insert
Product_Type.Times
Groups.minus_class.minus
Option.option.Some
AssocList.set
AssocList.delete
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
abbreviation UNIV :: "'a set"
where "UNIV \<equiv> top"
definition insert :: "'a \<Rightarrow> 'a set \<Rightar... | ###output
lemma set_delete: "set (delete k al) = set al - {k} \<times> UNIV"
###end |
Quantifier_Elimination_Hybrid/Hybrid_Multiv_Matrix_Proofs | Hybrid_Multiv_Matrix_Proofs.pull_out_pairs_length | lemma pull_out_pairs_length:
shows "length (pull_out_pairs qs Is) = length Is" | length (pull_out_pairs ?qs ?Is) = length ?Is | ?H1 (?H2 x_1 x_2) = ?H3 x_2 | lemma_command | ###symbols
Hybrid_Multiv_Algorithm.calculate_data_to_signs
Hybrid_Multiv_Matrix.construct_NofI_M
List.list.Cons
Renegar_Algorithm.combine_systems_R
List.length
Hybrid_Multiv_Matrix.pull_out_pairs
###defs
fun calculate_data_to_signs:: "(assumps \<times> matrix_equation) list \<Rightarrow> (assumps \<times> rat list list... | ###output
lemma pull_out_pairs_length:
shows "length (pull_out_pairs qs Is) = length Is"
###end |
Jinja/BV/BVSpecTypeSafe | BVSpecTypeSafe.progress_conform | lemma progress_conform:
"\<lbrakk>wf_jvm_prog\<^bsub>\<Phi>\<^esub> P; P,\<Phi> \<turnstile> (xp,h,frs)\<surd>; xp=None; frs\<noteq>[]\<rbrakk>
\<Longrightarrow> \<exists>\<sigma>'. P \<turnstile> (xp,h,frs) -jvm\<rightarrow>\<^sub>1 \<sigma>' \<and> P,\<Phi> \<turnstile> \<sigma>'\<surd>" | wf_jvm_prog\<^bsub>?\<Phi>\<^esub> ?P \<Longrightarrow> ?P,?\<Phi> |- (?xp, ?h, ?frs) [ok] \<Longrightarrow> ?xp = None \<Longrightarrow> ?frs \<noteq> [] \<Longrightarrow> \<exists>\<sigma>'. ?P \<turnstile> (?xp, ?h, ?frs) -jvm\<rightarrow>\<^sub>1 \<sigma>' \<and> ?P,?\<Phi> |- \<sigma>' [ok] | \<lbrakk>?H1 x_1 x_2; ?H2 x_2 x_1 (x_3, x_4, x_5); x_3 = ?H3; x_5 \<noteq> ?H4\<rbrakk> \<Longrightarrow> \<exists>y_0. ?H5 x_2 (x_3, x_4, x_5) y_0 \<and> ?H2 x_2 x_1 y_0 | lemma_command | ###symbols
BVSpec.wf_jvm_prog_phi
List.list.Nil
Option.option.None
JVMExec.exec_1'
BVConform.correct_state
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype 'a option =
None
| Some (th... | ###output
lemma progress_conform:
"\<lbrakk>wf_jvm_prog\<^bsub>\<Phi>\<^esub> P; P,\<Phi> \<turnstile> (xp,h,frs)\<surd>; xp=None; frs\<noteq>[]\<rbrakk>
\<Longrightarrow> \<exists>\<sigma>'. P \<turnstile> (xp,h,frs) -jvm\<rightarrow>\<^sub>1 \<sigma>' \<and> P,\<Phi> \<turnstile> \<sigma>'\<surd>"
###end |
JinjaThreads/DFA/Product | Product_Type.fst_apfst | null | fst (apfst ?f ?x) = ?f (fst ?x) | ?H1 (?H2 x_1 x_2) = x_1 (?H3 x_2) | lemma_command | ###symbols
Product.le
Product_Type.apfst
Product_Type.prod.fst
Product_Type.internal_case_prod
###defs
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
where "apfst f = map_prod f id"
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
definition in... | ###output
None
###end |
Computational_Algebra/Formal_Power_Series | Formal_Power_Series.radical_power | lemma radical_power:
assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
and a0: "(a$0 :: 'a::field_char_0) \<noteq> 0"
shows "(fps_radical r (Suc k) (a ^ Suc k)) = a" | ?r (Suc ?k) (fps_nth ?a 0 ^ Suc ?k) = fps_nth ?a 0 \<Longrightarrow> fps_nth ?a 0 \<noteq> (0::?'a) \<Longrightarrow> fps_radical ?r (Suc ?k) (?a ^ Suc ?k) = ?a | \<lbrakk>x_1 (?H1 x_2) (?H2 (?H3 x_3 ?H4) (?H1 x_2)) = ?H3 x_3 ?H4; ?H3 x_3 ?H4 \<noteq> ?H5\<rbrakk> \<Longrightarrow> ?H6 x_1 (?H1 x_2) (?H7 x_3 (?H1 x_2)) = x_3 | lemma_command | ###symbols
Groups.zero_class.zero
Formal_Power_Series.subdegree
Nat.Suc
Power.power_class.power
Formal_Power_Series.fps.fps_nth
Formal_Power_Series.fps_radical
###defs
class zero =
fixes zero :: 'a ("0")
definition subdegree :: "('a::zero) fps \<Rightarrow> nat" where
"subdegree f = (if f = 0 then 0 else LEAST n. ... | ###output
lemma radical_power:
assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
and a0: "(a$0 :: 'a::field_char_0) \<noteq> 0"
shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
###end |
Virtual_Substitution/ExecutiblePolyProps | ExecutiblePolyProps.coeff_zero | lemma coeff_zero[simp]: "MPoly_Type.coeff 0 x = 0" | MPoly_Type.coeff 0 ?x = (0::?'a) | ?H1 ?H2 x_1 = ?H3 | lemma_command | ###symbols
Groups.zero_class.zero
MPoly_Type.coeff
###defs
class zero =
fixes zero :: 'a ("0")
definition coeff :: "'a::zero mpoly \<Rightarrow> (nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'a"
where
"coeff p = Poly_Mapping.lookup (mapping_of p)"
| ###output
lemma coeff_zero[simp]: "MPoly_Type.coeff 0 x = 0"
###end |
Epistemic_Logic/Epistemic_Logic | Epistemic_Logic.S5_S5' | lemma S5_S5': \<open>AxTB4 \<turnstile> p \<Longrightarrow> AxT5 \<turnstile> p\<close> | AxTB4 \<turnstile> ?p \<Longrightarrow> AxT5 \<turnstile> ?p | ?H1 ?H2 x_1 \<Longrightarrow> ?H1 ?H3 x_1 | lemma_command | ###symbols
Epistemic_Logic.AxTB4
Epistemic_Logic.AK
Epistemic_Logic.AxT5
###defs
abbreviation AxTB4 :: \<open>'i fm \<Rightarrow> bool\<close> where
\<open>AxTB4 \<equiv> AxT \<oplus> AxB \<oplus> Ax4\<close>
inductive AK :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm \<Rightarrow> bool\<close> (\<open>_ \<... | ###output
lemma S5_S5': \<open>AxTB4 \<turnstile> p \<Longrightarrow> AxT5 \<turnstile> p\<close>
###end |
Library/Linear_Temporal_Logic_on_Streams | Linear_Temporal_Logic_on_Streams.until_not_relesased_now | lemma until_not_relesased_now: "(\<phi> until \<psi>) \<omega> \<Longrightarrow> \<not> \<psi> \<omega> \<Longrightarrow> \<phi> \<omega>" | (?\<phi> until ?\<psi>) ?\<omega> \<Longrightarrow> \<not> ?\<psi> ?\<omega> \<Longrightarrow> ?\<phi> ?\<omega> | \<lbrakk>?H1 x_1 x_2 x_3; \<not> x_2 x_3\<rbrakk> \<Longrightarrow> x_1 x_3 | lemma_command | ###symbols
Linear_Temporal_Logic_on_Streams.UNTIL
###defs
coinductive UNTIL (infix "until" 60) for \<phi> \<psi> where
base: "\<psi> xs \<Longrightarrow> (\<phi> until \<psi>) xs"
|
step: "\<lbrakk>\<phi> xs; (\<phi> until \<psi>) (stl xs)\<rbrakk> \<Longrightarrow> (\<phi> until \<psi>) xs"
| ###output
lemma until_not_relesased_now: "(\<phi> until \<psi>) \<omega> \<Longrightarrow> \<not> \<psi> \<omega> \<Longrightarrow> \<phi> \<omega>"
###end |
List-Infinite/CommonSet/SetIntervalStep | SetIntervalStep.iprev_inext | lemma iprev_inext: "
n \<noteq> Max I \<or> infinite I \<Longrightarrow> iprev (inext n I) I = n" | ?n \<noteq> Max ?I \<or> infinite ?I \<Longrightarrow> iprev (inext ?n ?I) ?I = ?n | x_1 \<noteq> ?H1 x_2 \<or> ?H2 x_2 \<Longrightarrow> ?H3 (?H4 x_1 x_2) x_2 = x_1 | lemma_command | ###symbols
Lattices_Big.linorder_class.Max
SetIntervalStep.inext
Finite_Set.infinite
SetIntervalStep.iprev
###defs
abbreviation infinite :: "'a set \<Rightarrow> bool"
where "infinite S \<equiv> \<not> finite S"
| ###output
lemma iprev_inext: "
n \<noteq> Max I \<or> infinite I \<Longrightarrow> iprev (inext n I) I = n"
###end |
Sort_Encodings/G | Groebner_Basis.bool_simps(29) | null | (?P \<or> True) = True | (x_1 \<or> True) = True | lemma_command | ###symbols
Relation.transp
M.Struct.satPB
Groups_Big.comm_monoid_mult_class.prod
G.ProblemIkTpartG.GE_parOf
###defs
abbreviation transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"transp \<equiv> transp_on UNIV"
| ###output
None
###end |
CoSMed/Friend_Confidentiality/Friend | Friend.iaction_mono | null | iaction ?\<Delta> ?s ?vl ?s1.0 ?vl1.0 \<Longrightarrow> (\<And>s vl s1 vl1. ?\<Delta> s vl s1 vl1 \<Longrightarrow> ?\<Delta>' s vl s1 vl1) \<Longrightarrow> iaction ?\<Delta>' ?s ?vl ?s1.0 ?vl1.0 | \<lbrakk>?H1 x_1 x_2 x_3 x_4 x_5; \<And>y_0 y_1 y_2 y_3. x_1 y_0 y_1 y_2 y_3 \<Longrightarrow> x_6 y_0 y_1 y_2 y_3\<rbrakk> \<Longrightarrow> ?H1 x_6 x_2 x_3 x_4 x_5 | lemma_command | ###symbols
Friend.iaction
###defs
| ###output
None
###end |
ex/Reflection_Examples | Reflection_Examples.Inum_number | lemma Inum_number: "Inum (C (numeral t)) vs = numeral t" | Inum (C (numeral ?t)) ?vs = numeral ?t | ?H1 (?H2 (?H3 x_1)) x_2 = ?H3 x_1 | lemma_command | ###symbols
Num.numeral_class.numeral
Reflection_Examples.Inum
Reflection_Examples.aform.Ge
Reflection_Examples.num.C
###defs
primrec numeral :: "num \<Rightarrow> 'a"
where
numeral_One: "numeral One = 1"
| numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
| numeral_Bit1: "numeral (Bit1 n) = numeral n +... | ###output
lemma Inum_number: "Inum (C (numeral t)) vs = numeral t"
###end |
Store_Buffer_Reduction/ReduceStoreBuffer | ReduceStoreBuffer.last_prog_hd_prog_append | lemma last_prog_hd_prog_append:
"last_prog (hd_prog p\<^sub>s\<^sub>b (sb@sb')) sb =last_prog (hd_prog p\<^sub>s\<^sub>b sb') sb" | last_prog (hd_prog ?p\<^sub>s\<^sub>b (?sb @ ?sb')) ?sb = last_prog (hd_prog ?p\<^sub>s\<^sub>b ?sb') ?sb | ?H1 (?H2 x_1 (?H3 x_2 x_3)) x_2 = ?H1 (?H2 x_1 x_3) x_2 | lemma_command | ###symbols
Relation.transp
ReduceStoreBuffer.last_prog
ReduceStoreBuffer.hd_prog
List.append
###defs
abbreviation transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"transp \<equiv> transp_on UNIV"
fun last_prog:: "'p \<Rightarrow> 'p store_buffer \<Rightarrow> 'p"
where
"last_prog p [] =... | ###output
lemma last_prog_hd_prog_append:
"last_prog (hd_prog p\<^sub>s\<^sub>b (sb@sb')) sb =last_prog (hd_prog p\<^sub>s\<^sub>b sb') sb"
###end |
Query_Optimization/Dtree | Dtree.dverts_subtree_subset | lemma dverts_subtree_subset: "is_subtree x y \<Longrightarrow> dverts x \<subseteq> dverts y" | is_subtree ?x ?y \<Longrightarrow> dverts ?x \<subseteq> dverts ?y | ?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_1) (?H3 x_2) | lemma_command | ###symbols
Dtree.dtree.dverts
Digraph.pre_digraph.verts
Dtree.wf_darcs'
Multiset.add_mset
Dtree.is_subtree
Dtree.dverts_mset_dom
Set.subset_eq
###defs
datatype (dverts:'a, darcs: 'b) dtree = Node (root: 'a) (sucs: "(('a,'b) dtree \<times> 'b) fset")
record ('a,'b) pre_digraph =
verts :: "'a set"
arcs :: "'b set"
... | ###output
lemma dverts_subtree_subset: "is_subtree x y \<Longrightarrow> dverts x \<subseteq> dverts y"
###end |
HOL-CSP/Process | Process.nil_less | lemma nil_less[simp]: "\<not> t < []" | \<not> ?t < [] | \<not> x_1 < ?H1 | lemma_command | ###symbols
Groups.minus_class.minus
List.list.Nil
List.list.tl
Finite_Set.card
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
wher... | ###output
lemma nil_less[simp]: "\<not> t < []"
###end |
Ordered_Resolution_Prover/Lazy_List_Chain | Lazy_List_Chain.emb_LConsE | null | emb (LCons ?z ?zs) ?ys \<Longrightarrow> (\<And>ys zs. ?ys = prepend zs (LCons ?z ys) \<Longrightarrow> emb ?zs ys \<Longrightarrow> ?P) \<Longrightarrow> ?P | \<lbrakk>?H1 (?H2 x_1 x_2) x_3; \<And>y_0 y_1. \<lbrakk>x_3 = ?H3 y_1 (?H2 x_1 y_0); ?H1 x_2 y_0\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4 | lemma_command | ###symbols
Coinductive_List.llist.lhd
Lazy_List_Chain.emb
List.list.case_list
Groups.zero_class.zero
Coinductive_List.llist.LCons
Lazy_List_Chain.chain
Lazy_List_Chain.prepend
###defs
codatatype (lset: 'a) llist =
lnull: LNil
| LCons (lhd: 'a) (ltl: "'a llist")
for
map: lmap
rel: llist_all2
where
"lhd LNil ... | ###output
None
###end |
Tycon/Maybe_Monad | Maybe_Monad.plusU_Nothing_right | lemma plusU_Nothing_right: "plusU\<cdot>xs\<cdot>Nothing = xs" | plusU\<cdot>?xs\<cdot>Nothing = ?xs | ?H1 (?H2 ?H3 x_1) ?H4 = x_1 | lemma_command | ###symbols
Tr.TT
Monad_Plus.plusU_class.plusU
Cfun.cfun.Rep_cfun
Maybe_Monad.maybe.Nothing
###defs
definition TT :: "tr"
where "TT = Def True"
class plusU = tycon +
fixes plusU :: "udom\<cdot>'a \<rightarrow> udom\<cdot>'a \<rightarrow> udom\<cdot>'a::tycon"
| ###output
lemma plusU_Nothing_right: "plusU\<cdot>xs\<cdot>Nothing = xs"
###end |
Deriving/Derive_Examples | Derive_Examples.comparator_mtree_pointwise(2) | null | (\<And>x\<^sub>0\<^sub>_\<^sub>0. x\<^sub>0\<^sub>_\<^sub>0 \<in> set_mtree ?x\<^sub>0 \<Longrightarrow> psym_comp ?comp\<^sub>'\<^sub>a x\<^sub>0\<^sub>_\<^sub>0) \<Longrightarrow> psym_comp (comparator_mtree ?comp\<^sub>'\<^sub>a) ?x\<^sub>0 | (\<And>y_0. y_0 \<in> ?H1 x_1 \<Longrightarrow> ?H2 x_2 y_0) \<Longrightarrow> ?H3 (?H4 x_2) x_1 | lemma_command | ###symbols
Derive_Examples.complex.map_complex
Comparator_Generator.psym_comp
Derive_Examples.mtree.set_mtree
Derive_Examples.complex2.size_complex2
HashCode.hashable_class.hashcode
Derive_Examples.comparator_mtree
###defs
definition psym_comp :: "'a comparator \<Rightarrow> 'a \<Rightarrow> bool" where
"psym_comp ac... | ###output
None
###end |
Decl_Sem_Fun_PL/DenotLam5 | DenotLam5.e_lam_intro | lemma e_lam_intro[intro]: "\<lbrakk> v = VFun f;
\<forall> v1 v2. (v1,v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x,v1)#\<rho>) \<rbrakk>
\<Longrightarrow> v \<in> E (ELam x e) \<rho>" | ?v = VFun ?f \<Longrightarrow> \<forall>v1 v2. (v1, v2) \<in> set ?f \<longrightarrow> v2 \<in> E ?e ((?x, v1) # ?\<rho>) \<Longrightarrow> ?v \<in> E (ELam ?x ?e) ?\<rho> | \<lbrakk>x_1 = ?H1 x_2; \<forall>y_0 y_1. (y_0, y_1) \<in> ?H2 x_2 \<longrightarrow> y_1 \<in> ?H3 x_3 (?H4 (x_4, y_0) x_5)\<rbrakk> \<Longrightarrow> x_1 \<in> ?H3 (?H5 x_4 x_3) x_5 | lemma_command | ###symbols
DeclSemAsDenot.E
List.list.Nil
Lambda.exp.ELam
List.list.set
Values.val.VFun
List.list.Cons
Lambda.FV
###defs
fun E :: "exp \<Rightarrow> env \<Rightarrow> val set" where
Enat: "E (ENat n) \<rho> = { v. v = VNat n }" |
Evar: "E (EVar x) \<rho> = { v. \<exists> v'. lookup \<rho> x = Some v' \<and> v \<sqs... | ###output
lemma e_lam_intro[intro]: "\<lbrakk> v = VFun f;
\<forall> v1 v2. (v1,v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x,v1)#\<rho>) \<rbrakk>
\<Longrightarrow> v \<in> E (ELam x e) \<rho>"
###end |
JinjaDCI/Compiler/TypeComp | TypeComp.wt_instr_append | lemma wt_instr_append:
assumes wti: "P,T,m,mpc - size \<tau>s',[] \<turnstile> i,pc - size \<tau>s' :: \<tau>s"
and pcl: "size \<tau>s' \<le> pc" and mpcl: "size \<tau>s' \<le> mpc"
and pcu: "pc < size \<tau>s + size \<tau>s'" and mpcu: "mpc \<le> size \<tau>s + size \<tau>s'"
shows "P,T,m,mpc,[] \<turnstile> i,pc ... | ?P,?T,?m,?mpc - length ?\<tau>s',[] \<turnstile> ?i,?pc - length ?\<tau>s' :: ?\<tau>s \<Longrightarrow> length ?\<tau>s' \<le> ?pc \<Longrightarrow> length ?\<tau>s' \<le> ?mpc \<Longrightarrow> ?pc < length ?\<tau>s + length ?\<tau>s' \<Longrightarrow> ?mpc \<le> length ?\<tau>s + length ?\<ta... | \<lbrakk>?H1 x_1 x_2 x_3 (?H2 x_4 (?H3 x_5)) ?H4 x_6 (?H2 x_7 (?H3 x_5)) x_8; ?H3 x_5 \<le> x_7; ?H3 x_5 \<le> x_4; x_7 < ?H5 (?H3 x_8) (?H3 x_5); x_4 \<le> ?H5 (?H3 x_8) (?H3 x_5)\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_3 x_4 ?H4 x_6 x_7 (?H6 x_5 x_8) | lemma_command | ###symbols
List.append
Groups.plus_class.plus
BVSpec.wt_instr
List.list.Nil
Groups.minus_class.minus
List.length
###defs
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
class plus =
fixes plus :: "'a \<... | ###output
lemma wt_instr_append:
assumes wti: "P,T,m,mpc - size \<tau>s',[] \<turnstile> i,pc - size \<tau>s' :: \<tau>s"
and pcl: "size \<tau>s' \<le> pc" and mpcl: "size \<tau>s' \<le> mpc"
and pcu: "pc < size \<tau>s + size \<tau>s'" and mpcu: "mpc \<le> size \<tau>s + size \<tau>s'"
shows "P,T,m,mpc,[] \<turnst... |
Factored_Transition_System_Bounding/FactoredSystem | FactoredSystem.submap_imp_state_succ_submap | lemma submap_imp_state_succ_submap:
fixes a :: "'a action" and s1 s2
assumes "(fst a \<subseteq>\<^sub>f s1)" "(s1 \<subseteq>\<^sub>f s2)"
shows "(state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a)" | fst ?a \<subseteq>\<^sub>f ?s1.0 \<Longrightarrow> ?s1.0 \<subseteq>\<^sub>f ?s2.0 \<Longrightarrow> state_succ ?s1.0 ?a \<subseteq>\<^sub>f state_succ ?s2.0 ?a | \<lbrakk>?H1 (?H2 x_1) x_2; ?H1 x_2 x_3\<rbrakk> \<Longrightarrow> ?H1 (?H3 x_2 x_1) (?H3 x_3 x_1) | lemma_command | ###symbols
Finite_Map.fmsubset
Product_Type.prod.fst
FactoredSystem.inj
FactoredSystemLib.state_succ
List.list.Cons
List.filter
Set.image
###defs
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
definition inj :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" whe... | ###output
lemma submap_imp_state_succ_submap:
fixes a :: "'a action" and s1 s2
assumes "(fst a \<subseteq>\<^sub>f s1)" "(s1 \<subseteq>\<^sub>f s2)"
shows "(state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a)"
###end |
UPF_Firewall/FWNormalisation/NormalisationIPPProofs | NormalisationIPPProofs.C_eqLemmas_id(19) | null | noDenyAll1 ?p \<Longrightarrow> NetsCollected2 (separate ?p) | ?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_1) | lemma_command | ###symbols
FWNormalisationCore.NetsCollected2
FWNormalisationCore.noDenyAll1
FWNormalisationCore.separate
###defs
| ###output
None
###end |
Complx/SeqCatch_decomp | SeqCatch_decomp.Seq_decomp_relpow | lemma Seq_decomp_relpow:
"\<Gamma> \<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>nn (p', Normal s') \<Longrightarrow>
final (p', Normal s') \<Longrightarrow>
(\<exists>n1<n. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Throw, Normal s')) \<and> p'=Throw \<or>
(\<exists>... | ?\<Gamma>\<turnstile> (Seq ?p\<^sub>1 ?p\<^sub>2, Normal ?s) \<rightarrow>\<^sup>n?n (?p', Normal ?s') \<Longrightarrow> final (?p', Normal ?s') \<Longrightarrow> (\<exists>n1<?n. ?\<Gamma>\<turnstile> (?p\<^sub>1, Normal ?s) \<rightarrow>\<^sup>nn1 (Throw, Normal ?s')) \<and> ?p' = Throw \<or> ... | \<lbrakk>?H1 x_1 (?H2 x_2 x_3, ?H3 x_4) x_5 (x_6, ?H3 x_7); ?H4 (x_6, ?H3 x_7)\<rbrakk> \<Longrightarrow> (\<exists>y_0<x_5. ?H1 x_1 (x_2, ?H3 x_4) y_0 (?H5, ?H3 x_7)) \<and> x_6 = ?H5 \<or> (\<exists>y_1 y_2 y_3. ?H1 x_1 (x_2, ?H3 x_4) y_... | lemma_command | ###symbols
SmallStep.final
SmallStep.step_n_trancl
SmallStep.step_rtrancl
Language.com.Throw
Language.com.Skip
SmallStep.xstate.Normal
Language.com.Seq
###defs
definition final:: "('s,'p,'f) config \<Rightarrow> bool" where
"final cfg = (fst cfg=Skip \<or> (fst cfg=Throw \<and> (\<exists>s. snd cfg=Normal s)))"
abbrevi... | ###output
lemma Seq_decomp_relpow:
"\<Gamma> \<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>nn (p', Normal s') \<Longrightarrow>
final (p', Normal s') \<Longrightarrow>
(\<exists>n1<n. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Throw, Normal s')) \<and> p'=Throw \<or>
... |
HOLCF/IOA/Automata | Automata.reachable_0 | null | ?s \<in> starts_of ?C \<Longrightarrow> reachable ?C ?s | x_1 \<in> ?H1 x_2 \<Longrightarrow> ?H2 x_2 x_1 | lemma_command | ###symbols
Option.option.Some
Automata.reachable
Automata.invariant
Automata.starts_of
###defs
datatype 'a option =
None
| Some (the: 'a)
| ###output
None
###end |
AWN/OInvariants | OInvariants.subreachableE_pair | lemma subreachableE_pair [elim]:
assumes "subreachable A U J"
and "(\<sigma>, s) \<in> oreachable A (\<lambda>s s'. I) U"
shows "\<exists>\<zeta>. (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<and> (\<zeta>, s) \<in> reachable A I" | subreachable ?A ?U ?J \<Longrightarrow> (?\<sigma>, ?s) \<in> oreachable ?A (\<lambda>s s'. ?I) ?U \<Longrightarrow> \<exists>\<zeta>. (\<forall>j\<in>?J. \<zeta> j = ?\<sigma> j) \<and> (\<zeta>, ?s) \<in> reachable ?A ?I | \<lbrakk>?H1 x_1 x_2 x_3; (x_4, x_5) \<in> ?H2 x_1 (\<lambda>y_0 y_1. x_6) x_2\<rbrakk> \<Longrightarrow> \<exists>y_2. (\<forall>y_3\<in>x_3. y_2 y_3 = x_4 y_3) \<and> (y_2, x_5) \<in> ?H3 x_1 x_6 | lemma_command | ###symbols
OInvariants.subreachable
OInvariants.otherwith
Product_Type.prod.fst
Product_Type.prod.snd
OInvariants.oreachable
OInvariants.local_steps
Invariants.reachable
###defs
definition subreachable
where "subreachable A U J \<equiv> \<forall>I. \<forall>s \<in> oreachable A (\<lambda>s s'. I) U.
... | ###output
lemma subreachableE_pair [elim]:
assumes "subreachable A U J"
and "(\<sigma>, s) \<in> oreachable A (\<lambda>s s'. I) U"
shows "\<exists>\<zeta>. (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<and> (\<zeta>, s) \<in> reachable A I"
###end |
QHLProver/Complex_Matrix | Complex_Matrix.positive_if_decomp | lemma positive_if_decomp:
assumes dimA: "A \<in> carrier_mat n n" and "\<exists>M. M * adjoint M = A"
shows "positive A" | ?A \<in> carrier_mat ?n ?n \<Longrightarrow> \<exists>M. M * adjoint M = ?A \<Longrightarrow> positive ?A | \<lbrakk>x_1 \<in> ?H1 x_2 x_2; \<exists>y_0. ?H2 y_0 (?H3 y_0) = x_1\<rbrakk> \<Longrightarrow> ?H4 x_1 | lemma_command | ###symbols
Complex_Matrix.unitary_schur_decomposition_dom
Matrix.carrier_mat
Complex_Matrix.adjoint
Complex_Matrix.vec_norm
Complex_Matrix.positive
Groups.times_class.times
###defs
definition carrier_mat :: "nat \<Rightarrow> nat \<Rightarrow> 'a mat set"
where "carrier_mat nr nc = { m . dim_row m = nr \<and> dim_col... | ###output
lemma positive_if_decomp:
assumes dimA: "A \<in> carrier_mat n n" and "\<exists>M. M * adjoint M = A"
shows "positive A"
###end |
Query_Optimization/Dtree | Dtree.sum_img_eq | lemma sum_img_eq:
assumes "\<forall>t \<in> fst ` fset xs. (g::'a \<Rightarrow> nat) (f t) = g t"
and "fcard ((\<lambda>(t,e). (f t, e)) |`| xs) = fcard xs"
shows "(\<Sum>(x,y)\<in> fset ((\<lambda>(t,e). (f t, e)) |`| xs). g x) = (\<Sum>(x,y)\<in> fset xs. g x)" | \<forall>t\<in>fst ` fset ?xs. ?g (?f t) = ?g t \<Longrightarrow> fcard ((\<lambda>(t, e). (?f t, e)) |`| ?xs) = fcard ?xs \<Longrightarrow> (\<Sum>(x, y)\<in>fset ((\<lambda>(t, e). (?f t, e)) |`| ?xs). ?g x) = (\<Sum>(x, y)\<in>fset ?xs. ?g x) | \<lbrakk>\<forall>y_0\<in>?H1 ?H2 (?H3 x_1). x_2 (x_3 y_0) = x_2 y_0; ?H4 (?H5 (?H6 (\<lambda>y_1. Pair (x_3 y_1))) x_1) = ?H4 x_1\<rbrakk> \<Longrightarrow> ?H7 (?H8 (\<lambda>y_3 y_4. x_2 y_3)) (?H3 (?H5 (?H6 (\<lambda>y_5. Pair (x_3 y_5))) x_1)) = ?H7 (?H8 (\<lambda>y_7 y_8. x_2... | lemma_command | ###symbols
Groups_Big.comm_monoid_add_class.sum
Product_Type.prod.fst
FSet.fset.fset
FSet.fcard
Set.image
FSet.fimage
Product_Type.prod.case_prod
###defs
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "... | ###output
lemma sum_img_eq:
assumes "\<forall>t \<in> fst ` fset xs. (g::'a \<Rightarrow> nat) (f t) = g t"
and "fcard ((\<lambda>(t,e). (f t, e)) |`| xs) = fcard xs"
shows "(\<Sum>(x,y)\<in> fset ((\<lambda>(t,e). (f t, e)) |`| xs). g x) = (\<Sum>(x,y)\<in> fset xs. g x)"
###end |
UNITY/SubstAx | SubstAx.leadsTo_imp_LeadsTo | lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B" | ?F \<in> ?A \<longmapsto> ?B \<Longrightarrow> ?F \<in> ?A \<longmapsto>w ?B | x_1 \<in> ?H1 x_2 x_3 \<Longrightarrow> x_1 \<in> ?H2 x_2 x_3 | lemma_command | ###symbols
Set.subset_eq
Constrains.Stable
Complete_Lattices.Union
SubstAx.LeadsTo
WFair.leadsTo
###defs
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
definition Stable :: "'a set => 'a program set" where
"Stable A == A Co A"
abbreviation Union :... | ###output
lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B"
###end |
Subresultants/Dichotomous_Lazard | Dichotomous_Lazard.dichotomous_Lazard | lemma dichotomous_Lazard: fixes x :: "'a :: factorial_ring_gcd"
assumes "(to_fract x)^n / (to_fract y)^(n-1) \<in> range to_fract"
shows "to_fract (dichotomous_Lazard x y n) = (to_fract x)^n / (to_fract y)^(n-1)" | to_fract ?x ^ ?n / to_fract ?y ^ (?n - 1) \<in> range to_fract \<Longrightarrow> to_fract (dichotomous_Lazard ?x ?y ?n) = to_fract ?x ^ ?n / to_fract ?y ^ (?n - 1) | ?H1 (?H2 (?H3 x_1) x_2) (?H2 (?H3 x_3) (?H4 x_2 ?H5)) \<in> ?H6 ?H3 \<Longrightarrow> ?H3 (?H7 x_1 x_3 x_2) = ?H1 (?H2 (?H3 x_1) x_2) (?H2 (?H3 x_3) (?H4 x_2 ?H5)) | lemma_command | ###symbols
Dichotomous_Lazard.dichotomous_Lazard
Groups.minus_class.minus
Groups.one_class.one
Polynomial_Factorial.to_fract
Fields.inverse_class.inverse_divide
Power.power_class.power
Set.range
###defs
fun dichotomous_Lazard :: "'a :: idom_divide \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" where
"dichotomou... | ###output
lemma dichotomous_Lazard: fixes x :: "'a :: factorial_ring_gcd"
assumes "(to_fract x)^n / (to_fract y)^(n-1) \<in> range to_fract"
shows "to_fract (dichotomous_Lazard x y n) = (to_fract x)^n / (to_fract y)^(n-1)"
###end |
Nonstandard_Analysis/Star | StarDef.starP_star_of | null | (*p* ?P) (star_of ?x) = ?P ?x | ?H1 x_1 (?H2 x_2) = x_1 x_2 | lemma_command | ###symbols
Set.Collect
Set.subset_eq
StarDef.FreeUltrafilterNat
StarDef.star_of
StarDef.starP
###defs
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
definition FreeUltrafilterNat :: "nat filter" (\<open>\<U>\<close>)
where "\<U> = (SOME U. freeultrafil... | ###output
None
###end |
Shivers-CFA/AbsCF | AbsCF.cont2cont_case_call | lemma cont2cont_case_call [simp, cont2cont]:
assumes "\<And>a b c. cont (\<lambda>x. f1 x a b c)"
and "\<And>a b c. cont (\<lambda>x. f2 x a b c)"
shows "cont (\<lambda>x. case_call (f1 x) (f2 x) c)" | (\<And>a b c. cont (\<lambda>x. ?f1.0 x a b c)) \<Longrightarrow> (\<And>a b c. cont (\<lambda>x. ?f2.0 x a b c)) \<Longrightarrow> cont (\<lambda>x. case ?c of App xa xb xc \<Rightarrow> ?f1.0 x xa xb xc | call.Let xa xb xc \<Rightarrow> ?f2.0 x xa xb xc) | \<lbrakk>\<And>y_0 y_1 y_2. ?H1 (\<lambda>y_3. x_1 y_3 y_0 y_1 y_2); \<And>y_4 y_5 y_6. ?H1 (\<lambda>y_7. x_2 y_7 y_4 y_5 y_6)\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_8. ?H2 (x_1 y_8) (x_2 y_8) x_3) | lemma_command | ###symbols
Cont.cont
CPSScheme.call.case_call
###defs
definition cont :: "('a::cpo \<Rightarrow> 'b::cpo) \<Rightarrow> bool"
where "cont f = (\<forall>Y. chain Y \<longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i))"
| ###output
lemma cont2cont_case_call [simp, cont2cont]:
assumes "\<And>a b c. cont (\<lambda>x. f1 x a b c)"
and "\<And>a b c. cont (\<lambda>x. f2 x a b c)"
shows "cont (\<lambda>x. case_call (f1 x) (f2 x) c)"
###end |
Nominal/Nominal | Nominal.div_nat_eqvt | lemma div_nat_eqvt:
fixes x::"nat"
shows "pi\<bullet>(x div y) = (pi\<bullet>x) div (pi\<bullet>y)" | ?pi \<bullet> (?x div ?y) = ?pi \<bullet> ?x div ?pi \<bullet> ?y | ?H1 x_1 (?H2 x_2 x_3) = ?H2 (?H1 x_1 x_2) (?H1 x_1 x_3) | lemma_command | ###symbols
Set.insert
Rings.divide_class.divide
BNF_Def.rel_fun
Nominal.perm
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
class divide =
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
definition
... | ###output
lemma div_nat_eqvt:
fixes x::"nat"
shows "pi\<bullet>(x div y) = (pi\<bullet>x) div (pi\<bullet>y)"
###end |
Taylor_Models/Polynomial_Expression | Polynomial_Expression.polynate_norm | lemma polynate_norm[simp]:
fixes p :: "'a::field poly"
shows "isnpoly (polynate p)" | isnpoly (polynate ?p) | ?H1 (?H2 x_1) | lemma_command | ###symbols
List.list.set
Polynomial_Expression.isnpoly
Polynomial_Expression.polynate
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
definition isnpoly :: "'a::zero poly \<Rightarrow> bool"
wher... | ###output
lemma polynate_norm[simp]:
fixes p :: "'a::field poly"
shows "isnpoly (polynate p)"
###end |
Probability/Distributions | Distributions.integrable_std_normal_moment_abs | lemma integrable_std_normal_moment_abs: "integrable lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar>^k)" | integrable lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar> ^ ?k) | ?H1 ?H2 (\<lambda>y_0. ?H3 (?H4 y_0) (?H5 (?H6 y_0) x_1)) | lemma_command | ###symbols
Power.power_class.power
Groups.abs_class.abs
Set.image
Distributions.std_normal_density
Sigma_Algebra.space
Groups.times_class.times
Information.information_space
Factorial.semiring_char_0_class.fact
Lebesgue_Measure.lborel
Bochner_Integration.integrable
###defs
primrec power :: "'a \<Rightarrow> nat \<Right... | ###output
lemma integrable_std_normal_moment_abs: "integrable lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar>^k)"
###end |
Order_Lattice_Props/Order_Lattice_Props | Order_Lattice_Props.fSup_distr | lemma fSup_distr: "Sup_pres (\<lambda>x. x \<circ> f)" | Sup_pres (\<lambda>x. x \<circ> ?f) | ?H1 (\<lambda>y_0. ?H2 y_0 x_1) | lemma_command | ###symbols
Fun.order_class.mono
Fun.comp
Order_Lattice_Props.Sup_pres
###defs
definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55)
where "f \<circ> g = (\<lambda>x. f (g x))"
abbreviation Sup_pres :: "('a::Sup \<Rightarrow> 'b::Sup) \<... | ###output
lemma fSup_distr: "Sup_pres (\<lambda>x. x \<circ> f)"
###end |
Stable_Matching/Choice_Functions | Choice_Functions.Aizerman_onI | null | (\<And>B C. B \<subseteq> ?A \<Longrightarrow> C \<subseteq> B \<Longrightarrow> ?f B \<subseteq> C \<Longrightarrow> ?f C \<subseteq> ?f B) \<Longrightarrow> Aizerman_on ?A ?f | (\<And>y_0 y_1. \<lbrakk>?H1 y_0 x_1; ?H1 y_1 y_0; ?H1 (x_2 y_0) y_1\<rbrakk> \<Longrightarrow> ?H1 (x_2 y_1) (x_2 y_0)) \<Longrightarrow> ?H2 x_1 x_2 | lemma_command | ###symbols
Set.subset_eq
Relation.trans
Choice_Functions.Aizerman_on
###defs
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
abbreviation trans :: "'a rel \<Rightarrow> bool" where
"trans \<equiv> trans_on UNIV"
definition Aizerman_on :: "'a set \<Righta... | ###output
None
###end |
Predicate_Compile_Examples/Predicate_Compile_Tests | Predicate_Compile_Tests.revP_i_iI | null | revP ?x ?xa \<Longrightarrow> pred.eval (revP_i_i ?x ?xa) () | ?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_1 x_2) ?H4 | lemma_command | ###symbols
Product_Type.Unity
Predicate.pred.eval
Predicate_Compile_Tests.revP_i_i
Predicate_Compile_Tests.compP
Predicate_Compile_Tests.nested_tuples_PPooio
Predicate_Compile_Tests.revP
Predicate_Compile_Tests.dseq_partition_FiBB_i_i_i
Predicate_Compile_Tests.uptP_i_i_i
###defs
definition Unity :: unit ("'(')")
whe... | ###output
None
###end |
Bali/TypeSafe | TypeSafe.error_free_sxalloc | lemma error_free_sxalloc:
assumes sxalloc: "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1" and error_free_s0: "error_free s0"
shows "error_free s1" | ?G\<turnstile>?s0.0 \<midarrow>sxalloc\<rightarrow> ?s1.0 \<Longrightarrow> error_free ?s0.0 \<Longrightarrow> error_free ?s1.0 | \<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H2 x_3 | lemma_command | ###symbols
Decl.field.type
Name.lname.case_lname
State.error_free
Eval.sxalloc
###defs
record field = member +
type :: ty
datatype lname \<comment> \<open>names for local variables and the This pointer\<close>
= EName ename
| This
definition
error_free :: "state \<Rightarrow> bool"
w... | ###output
lemma error_free_sxalloc:
assumes sxalloc: "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1" and error_free_s0: "error_free s0"
shows "error_free s1"
###end |
Rat | Rat.zero_le_Fract_iff | lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a" | 0 < ?b \<Longrightarrow> (0 \<le> Fract ?a ?b) = (0 \<le> ?a) | ?H1 < x_1 \<Longrightarrow> (?H2 \<le> ?H3 x_2 x_1) = (?H1 \<le> x_2) | lemma_command | ###symbols
Rat.Fract
Groups.zero_class.zero
Groups.uminus_class.uminus
###defs
class zero =
fixes zero :: 'a ("0")
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
| ###output
lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a"
###end |
Query_Optimization/Dtree | Dtree.darcs_mset_sub_darcs | lemma darcs_mset_sub_darcs: "set_mset (darcs_mset t) \<subseteq> darcs t" | set_mset (darcs_mset ?t) \<subseteq> darcs ?t | ?H1 (?H2 (?H3 x_1)) (?H4 x_1) | lemma_command | ###symbols
Dtree.dtree.pred_dtree
Multiset.set_mset
Dtree.insert_between_dom
Set.subset_eq
FSet.fempty
Relation.conversep
Dtree.darcs_mset
Dtree.dtree.darcs
###defs
datatype (dverts:'a, darcs: 'b) dtree = Node (root: 'a) (sucs: "(('a,'b) dtree \<times> 'b) fset")
definition set_mset :: \<open>'a multiset \<Rightarrow> ... | ###output
lemma darcs_mset_sub_darcs: "set_mset (darcs_mset t) \<subseteq> darcs t"
###end |
DFS_Framework/Examples/Nested_DFS | Nested_DFS.gbs_simps(15) | null | gbs_get_pending_update ?gbs_get_pending' (param_DFS_defs.gbs ?G) = \<lparr>gbs_init = RETURN \<circ> empty_state, gbs_is_empty_stack = is_empty_stack, gbs_new_root = (RETURN \<circ>\<circ>\<circ> param_DFS_defs.new_root) ?G, gbs_get_pending = ?gbs_get_pending' get_pending, gbs_finish = ... | ?H1 x_1 (?H2 x_2) = ?H3 (?H4 ?H5 ?H6) ?H7 (?H8 ?H5 ?H9 x_2) (x_1 ?H10) (?H11 ?H5 ?H12) ?H13 ?H14 (?H15 ?H5 ?H16) (?H15 ?H5 ?H17) (?H15 ?H5 (?H18 x_2)) ?H19 | lemma_command | ###symbols
Param_DFS.param_DFS_defs.discover
Misc.comp3
Fun.comp
General_DFS_Structure.param_DFS_defs.gbs
Nested_DFS.get_pending
General_DFS_Structure.gen_basic_dfs_struct.gbs_get_pending_update
Relators.option_rel
Nested_DFS.nested_dfs_code
Nested_DFS.is_finished
General_DFS_Structure.gen_basic_dfs_struct.gen_basic_df... | ###output
None
###end |
Nominal/Examples/W | W.fresh_atm(2) | null | ?a \<sharp> ?b = (?a \<noteq> ?b) | ?H1 x_1 x_2 = (x_1 \<noteq> x_2) | lemma_command | ###symbols
Nominal.fresh
###defs
definition fresh :: "'x \<Rightarrow> 'a \<Rightarrow> bool" (\<open>_ \<sharp> _\<close> [80,80] 80) where
"a \<sharp> x \<longleftrightarrow> a \<notin> supp x"
| ###output
None
###end |
Abstract-Rewriting/Abstract_Rewriting | Abstract_Rewriting.CR_imp_conversionIff_join | lemma CR_imp_conversionIff_join:
assumes "CR r" shows "r\<^sup>\<leftrightarrow>\<^sup>* = r\<^sup>\<down>" | CR ?r \<Longrightarrow> ?r\<^sup>\<leftrightarrow>\<^sup>* = ?r\<^sup>\<down> | ?H1 x_1 \<Longrightarrow> ?H2 x_1 = ?H3 x_1 | lemma_command | ###symbols
Abstract_Rewriting.conversion
Abstract_Rewriting.CR
Abstract_Rewriting.join
###defs
definition conversion :: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>\<leftrightarrow>\<^sup>*)" [1000] 999) where
"A\<^sup>\<leftrightarrow>\<^sup>* = (A\<^sup>\<leftrightarrow>)\<^sup>*"
abbreviation CR :: "'a rel \<Rightar... | ###output
lemma CR_imp_conversionIff_join:
assumes "CR r" shows "r\<^sup>\<leftrightarrow>\<^sup>* = r\<^sup>\<down>"
###end |
Security_Protocol_Refinement/Key_establish/m3_ds | m3_ds.m3_inv3_sesK_comprE | null | ?x \<in> m3_inv3_sesK_compr \<Longrightarrow> ((\<And>K KK. KK \<subseteq> range sesK \<Longrightarrow> (Key K \<in> analz (Key ` KK \<union> IK ?x)) = (K \<in> KK \<or> Key K \<in> analz (IK ?x))) \<Longrightarrow> PROP ?W) \<Longrightarrow> PROP ?W | \<lbrakk>x_1 \<in> ?H1; (\<And>y_0 y_1. ?H2 y_1 (?H3 ?H4) \<Longrightarrow> (?H5 y_0 \<in> ?H6 (?H7 (?H8 ?H5 y_1) (?H9 x_1))) = (y_0 \<in> y_1 \<or> ?H5 y_0 \<in> ?H6 (?H9 x_1))) \<Longrightarrow> PROP x_2\<rbrakk> \<Longrightarrow> PROP x_2 | lemma_command | ###symbols
Keys.key.sesK
Message.msg.Key
m3_ds.m3_inv3_sesK_compr
Message.analz
Set.range
Set.union
Set.image
Set.subset_eq
m3_ds.m3_state.IK
###defs
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
... | ###output
None
###end |
Factored_Transition_System_Bounding/FactoredSystem | FactoredSystem.lemma_1_i | lemma lemma_1_i:
fixes s a PROB
assumes "s \<in> valid_states PROB" "a \<in> PROB"
shows "state_succ s a \<in> valid_states PROB" | ?s \<in> valid_states ?PROB \<Longrightarrow> ?a \<in> ?PROB \<Longrightarrow> state_succ ?s ?a \<in> valid_states ?PROB | \<lbrakk>x_1 \<in> ?H1 x_2; x_3 \<in> x_2\<rbrakk> \<Longrightarrow> ?H2 x_1 x_3 \<in> ?H1 x_2 | lemma_command | ###symbols
List.append
Groups.minus_class.minus
FactoredSystemLib.valid_states
FactoredSystem.stateSpace
FactoredSystemLib.state_succ
###defs
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
class minus =
... | ###output
lemma lemma_1_i:
fixes s a PROB
assumes "s \<in> valid_states PROB" "a \<in> PROB"
shows "state_succ s a \<in> valid_states PROB"
###end |
Network_Security_Policy_Verification/Security_Invariants/SINVAR_BLPbasic | SINVAR_BLPbasic.ENFnrSR_to_ENFsr | null | sinvar_all_edges_normal_form_not_refl_SR ?P \<Longrightarrow> sinvar_all_edges_normal_form_sr (\<lambda>p1 v1 p2 v2. v1 \<noteq> v2 \<longrightarrow> ?P p1 v1 p2 v2) | ?H1 x_1 \<Longrightarrow> ?H2 (\<lambda>y_0 y_1 y_2 y_3. y_1 \<noteq> y_3 \<longrightarrow> x_1 y_0 y_1 y_2 y_3) | lemma_command | ###symbols
SINVAR_BLPbasic.sinvar_all_edges_normal_form_sr
SINVAR_BLPbasic.sinvar_all_edges_normal_form_not_refl_SR
###defs
| ###output
None
###end |
Library/Finite_Map | Finite_Map.fmfilter_add_distrib | lemma fmfilter_add_distrib[simp]: "fmfilter P (m ++\<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n" | fmfilter ?P (?m ++\<^sub>f ?n) = fmfilter ?P ?m ++\<^sub>f fmfilter ?P ?n | ?H1 x_1 (?H2 x_2 x_3) = ?H2 (?H1 x_1 x_2) (?H1 x_1 x_3) | lemma_command | ###symbols
Finite_Map.fmfilter
Finite_Map.fmadd
###defs
| ###output
lemma fmfilter_add_distrib[simp]: "fmfilter P (m ++\<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"
###end |
Computational_Algebra/Polynomial | Polynomial.coeffs_map_poly' | lemma coeffs_map_poly':
assumes "\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0"
shows "coeffs (map_poly f p) = map f (coeffs p)" | (\<And>x. x \<noteq> (0::?'a) \<Longrightarrow> ?f x \<noteq> (0::?'b)) \<Longrightarrow> coeffs (map_poly ?f ?p) = map ?f (coeffs ?p) | (\<And>y_0. y_0 \<noteq> ?H1 \<Longrightarrow> x_1 y_0 \<noteq> ?H2) \<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 x_1 (?H6 x_2) | lemma_command | ###symbols
Polynomial.map_poly
List.list.map
Polynomial.coeffs
Groups.zero_class.zero
###defs
definition map_poly :: "('a :: zero \<Rightarrow> 'b :: zero) \<Rightarrow> 'a poly \<Rightarrow> 'b poly"
where "map_poly f p = Poly (map f (coeffs p))"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a l... | ###output
lemma coeffs_map_poly':
assumes "\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0"
shows "coeffs (map_poly f p) = map f (coeffs p)"
###end |
Query_Optimization/Selectivities | Selectivities.ldeep_s_pos | lemma ldeep_s_pos: "sel_reasonable f \<Longrightarrow> ldeep_s f xs x > 0" | sel_reasonable ?f \<Longrightarrow> 0 < ldeep_s ?f ?xs ?x | ?H1 x_1 \<Longrightarrow> ?H2 < ?H3 x_1 x_2 x_3 | lemma_command | ###symbols
Selectivities.ldeep_s
Groups.zero_class.zero
Selectivities.sel_reasonable
###defs
fun ldeep_s :: "'a selectivity \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> real" where
"ldeep_s f [] = (\<lambda>_. 1)"
| "ldeep_s f (x#xs) = (\<lambda>a. if a=x then list_sel_aux' f xs a else ldeep_s f xs a)"
class ... | ###output
lemma ldeep_s_pos: "sel_reasonable f \<Longrightarrow> ldeep_s f xs x > 0"
###end |
MiniSail/Nominal-Utils | Nominal-Utils.fresh_prod5 | lemma fresh_prod5[nominal_prod_simps,ms_fresh]: "x \<sharp> (a,b,c,d,e) = (x \<sharp> a \<and> x \<sharp> b \<and> x \<sharp> c \<and> x \<sharp> d \<and> x \<sharp> e)" | ?x \<sharp> (?a, ?b, ?c, ?d, ?e) = (?x \<sharp> ?a \<and> ?x \<sharp> ?b \<and> ?x \<sharp> ?c \<and> ?x \<sharp> ?d \<and> ?x \<sharp> ?e) | ?H1 x_1 (x_2, x_3, x_4, x_5, x_6) = (?H2 x_1 x_2 \<and> ?H3 x_1 x_3 \<and> ?H4 x_1 x_4 \<and> ?H5 x_1 x_5 \<and> ?H6 x_1 x_6) | lemma_command | ###symbols
Sum_Type.sum.projl
Set.union
Set.range
Nominal2_Base.pt_class.fresh
###defs
definition "sum = {f. (\<exists>a. f = Inl_Rep (a::'a)) \<or> (\<exists>b. f = Inr_Rep (b::'b))}"
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
abbreviat... | ###output
lemma fresh_prod5[nominal_prod_simps,ms_fresh]: "x \<sharp> (a,b,c,d,e) = (x \<sharp> a \<and> x \<sharp> b \<and> x \<sharp> c \<and> x \<sharp> d \<and> x \<sharp> e)"
###end |
Core_DOM/common/preliminaries/Heap_Error_Monad | Heap_Error_Monad.bind_returns_error_eq | lemma bind_returns_error_eq:
assumes "h \<turnstile> f \<rightarrow>\<^sub>e e"
and "h \<turnstile> g \<rightarrow>\<^sub>e e"
shows "h \<turnstile> f = h \<turnstile> g" | ?h \<turnstile> ?f \<rightarrow>\<^sub>e ?e \<Longrightarrow> ?h \<turnstile> ?g \<rightarrow>\<^sub>e ?e \<Longrightarrow> ?h \<turnstile> ?f = ?h \<turnstile> ?g | \<lbrakk>?H1 x_1 x_2 x_3; ?H1 x_1 x_4 x_3\<rbrakk> \<Longrightarrow> ?H2 x_1 x_2 = ?H2 x_1 x_4 | lemma_command | ###symbols
Heap_Error_Monad.prog.case_prog
Heap_Error_Monad.returns_error
Sum_Type.Inr
Heap_Error_Monad.execute
Heap_Error_Monad.prog.the_prog
###defs
definition Inr :: "'b \<Rightarrow> 'a + 'b"
where "Inr = Abs_sum \<circ> Inr_Rep"
| ###output
lemma bind_returns_error_eq:
assumes "h \<turnstile> f \<rightarrow>\<^sub>e e"
and "h \<turnstile> g \<rightarrow>\<^sub>e e"
shows "h \<turnstile> f = h \<turnstile> g"
###end |
Flyspeck-Tame/Invariants | Invariants.minGraphProps6 | lemma minGraphProps6:
"minGraphProps g \<Longrightarrow> v : \<V> g \<Longrightarrow> f \<in> set (facesAt g v) \<Longrightarrow> v \<in> \<V> f" | minGraphProps ?g \<Longrightarrow> ?v \<in> \<V> ?g \<Longrightarrow> ?f \<in> set (facesAt ?g ?v) \<Longrightarrow> ?v \<in> \<V> ?f | \<lbrakk>?H1 x_1; x_2 \<in> ?H2 x_1; x_3 \<in> ?H3 (?H4 x_1 x_2)\<rbrakk> \<Longrightarrow> x_2 \<in> ?H5 x_3 | lemma_command | ###symbols
Graph.facesAt
List.list.set
Graph.face.Face
Graph.vertices_set
Invariants.minGraphProps
###defs
definition facesAt :: "graph \<Rightarrow> vertex \<Rightarrow> face list" where
"facesAt g v \<equiv> \<^cancel>\<open>if v \<in> set(vertices g) then\<close> faceListAt g ! v \<^cancel>\<open>else []\<close>"
d... | ###output
lemma minGraphProps6:
"minGraphProps g \<Longrightarrow> v : \<V> g \<Longrightarrow> f \<in> set (facesAt g v) \<Longrightarrow> v \<in> \<V> f"
###end |
Recursion-Theory-I/RecEnSet | RecEnSet.c_graph_lm_4 | lemma c_graph_lm_4: "c_graph f = ce_rel_to_set (graph f)" | c_graph ?f = ce_rel_to_set (graph ?f) | ?H1 x_1 = ?H2 (?H3 x_1) | lemma_command | ###symbols
RecEnSet.index_set
RecEnSet.graph
Relation.Image
RecEnSet.ce_rel_to_set
Groups.one_class.one
RecEnSet.c_graph
###defs
definition
index_set :: "nat set \<Rightarrow> bool" where
"index_set = (\<lambda> A. \<forall> n m. n \<in> A \<and> (nat_to_ce_set n = nat_to_ce_set m) \<longrightarrow> m \<in> A)"
def... | ###output
lemma c_graph_lm_4: "c_graph f = ce_rel_to_set (graph f)"
###end |
Gauss_Jordan/Matrix_To_IArray | Matrix_To_IArray.matrix_to_iarray_morph | lemma matrix_to_iarray_morph:
fixes A::"'a^'n::{mod_type}^'m::{mod_type}"
shows "(A = B) = (matrix_to_iarray A = matrix_to_iarray B)" | (?A = ?B) = (matrix_to_iarray ?A = matrix_to_iarray ?B) | (x_1 = x_2) = (?H1 x_1 = ?H1 x_2) | lemma_command | ###symbols
Matrix_To_IArray.column_iarray
Matrix_To_IArray.matrix_to_iarray
###defs
definition column_iarray :: "nat => 'a iarray iarray => 'a iarray"
where "column_iarray k A = IArray.of_fun (\<lambda>m. A !! m !! k) (IArray.length A)"
definition matrix_to_iarray :: "'a^'n::{mod_type}^'m::{mod_type} => 'a iarray ia... | ###output
lemma matrix_to_iarray_morph:
fixes A::"'a^'n::{mod_type}^'m::{mod_type}"
shows "(A = B) = (matrix_to_iarray A = matrix_to_iarray B)"
###end |
FO_Theory_Rewriting/Util/Ground_MCtxt | Ground_MCtxt.inf_gmctxt_comm | lemma inf_gmctxt_comm [ac_simps]:
"(C :: 'f gmctxt) \<sqinter> D = D \<sqinter> C" | ?C \<sqinter> ?D = ?D \<sqinter> ?C | ?H1 x_1 x_2 = ?H1 x_2 x_1 | lemma_command | ###symbols
Lattices.inf_class.inf
Ground_MCtxt.comp_gmctxtp
Ground_Ctxt.gctxt.GHole
Set.empty
Ground_MCtxt.gmctxt.pred_gmctxt
###defs
class inf =
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
| ###output
lemma inf_gmctxt_comm [ac_simps]:
"(C :: 'f gmctxt) \<sqinter> D = D \<sqinter> C"
###end |
Library/Linear_Temporal_Logic_on_Streams | Linear_Temporal_Logic_on_Streams.alw_ev_stl | lemma alw_ev_stl: "alw (ev P) (stl \<omega>) \<longleftrightarrow> alw (ev P) \<omega>" | alw (ev ?P) (stl ?\<omega>) = alw (ev ?P) ?\<omega> | ?H1 (?H2 x_1) (?H3 x_2) = ?H1 (?H2 x_1) x_2 | lemma_command | ###symbols
Linear_Temporal_Logic_on_Streams.HLD
Linear_Temporal_Logic_on_Streams.ev
Linear_Temporal_Logic_on_Streams.alw
Stream.stream.stl
###defs
definition "HLD s = holds (\<lambda>x. x \<in> s)"
inductive ev for \<phi> where
base: "\<phi> xs \<Longrightarrow> ev \<phi> xs"
|
step: "ev \<phi> (stl xs) \<Longrightarro... | ###output
lemma alw_ev_stl: "alw (ev P) (stl \<omega>) \<longleftrightarrow> alw (ev P) \<omega>"
###end |
Collections/ICF/CollectionsV1 | CollectionsV1.map_reverse_iterateoi_I | lemma map_reverse_iterateoi_I :
assumes "\<And>m. invar m \<Longrightarrow> map_iterator_rev_linord (ritoi m) (\<alpha> m)"
shows "map_reverse_iterateoi \<alpha> invar ritoi" | (\<And>m. ?invar m \<Longrightarrow> map_iterator_rev_linord (?ritoi m) (?\<alpha> m)) \<Longrightarrow> map_reverse_iterateoi ?\<alpha> ?invar ?ritoi | (\<And>y_0. x_1 y_0 \<Longrightarrow> ?H1 (x_2 y_0) (x_3 y_0)) \<Longrightarrow> ?H2 x_3 x_1 x_2 | lemma_command | ###symbols
HashSet.hs.\<alpha>
CollectionsV1.map_reverse_iterateoi
TrieMapImpl.tm.to_list
SetIterator.linorder_class.map_iterator_rev_linord
###defs
| ###output
lemma map_reverse_iterateoi_I :
assumes "\<And>m. invar m \<Longrightarrow> map_iterator_rev_linord (ritoi m) (\<alpha> m)"
shows "map_reverse_iterateoi \<alpha> invar ritoi"
###end |
Nat-Interval-Logic/IL_Interval | IL_Interval.iMOD_iMODb_inext | lemma iMOD_iMODb_inext: "
a < r + m * c \<Longrightarrow> inext a [r, mod m, c] = inext a [r, mod m]" | ?a < ?r + ?m * ?c \<Longrightarrow> inext ?a [ ?r, mod ?m, ?c ] = inext ?a [ ?r, mod ?m ] | x_1 < ?H1 x_2 (?H2 x_3 x_4) \<Longrightarrow> ?H3 x_1 (?H4 x_2 x_3 x_4) = ?H3 x_1 (?H5 x_2 x_3) | lemma_command | ###symbols
Groups.times_class.times
SetInterval2.iMax
Rings.divide_class.divide
IL_Interval.iMOD
IL_Interval.iMODb
Groups.plus_class.plus
SetIntervalStep.inext
Set.range
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class divide =
fixes divide :: "'a \<Rightarrow> 'a \... | ###output
lemma iMOD_iMODb_inext: "
a < r + m * c \<Longrightarrow> inext a [r, mod m, c] = inext a [r, mod m]"
###end |
AODV/variants/c_gtobcast/C_Aodv | C_Aodv_Data.update_cases | null | (\<pi>\<^sub>2 ?r = 0) = (\<pi>\<^sub>3 ?r = unk) \<Longrightarrow> (?ip \<notin> kD ?rt \<Longrightarrow> ?P (?rt(?ip \<mapsto> ?r))) \<Longrightarrow> (?ip \<in> kD ?rt \<Longrightarrow> sqn ?rt ?ip < \<pi>\<^sub>2 ?r \<Longrightarrow> ?P (?rt(?ip \<mapsto> ?r))) \<Longrightarrow> (?ip \<in> kD ?rt ... | \<lbrakk>(?H1 x_1 = ?H2) = (?H3 x_1 = ?H4); ?H5 x_2 (?H6 x_3) \<Longrightarrow> x_4 (?H7 x_3 x_2 (?H8 x_1)); \<lbrakk>x_2 \<in> ?H6 x_3; ?H9 x_3 x_2 < ?H1 x_1\<rbrakk> \<Longrightarrow> x_4 (?H7 x_3 x_2 (?H8 x_1)); \<lbrakk>x_2 \<in> ?H6 x_3; ?H9 x_3 x_2 = ?H1 x_1; ?H10 x_1 < ?H11 (?H12 x_3 x_2)\<rbrakk> \<Longr... | lemma_command | ###symbols
C_Aodv.state.data
Fun.fun_upd
Aodv_Basic.kno
Option.option.Some
Set.not_member
Groups.zero_class.zero
C_Aodv_Data.dhops
Aodv_Basic.inv
Option.option.the
C_Aodv_Data.proj2
C_Aodv_Data.proj3
C_Aodv.is_rrep
Aodv_Basic.val
C_Aodv_Data.kD
C_Aodv_Data.flag
C_Aodv_Data.proj6
Aodv_Basic.unk
C_Aodv_Data.proj5
C_Aodv_... | ###output
None
###end |
Stateful_Protocol_Composition_and_Typing/More_Unification | More_Unification.subst_Var_notin_img | lemma subst_Var_notin_img: "x \<notin> range_vars s \<Longrightarrow> t \<cdot> s = Var x \<Longrightarrow> t = Var x" | ?x \<notin> range_vars ?s \<Longrightarrow> ?t \<cdot> ?s = Var ?x \<Longrightarrow> ?t = Var ?x | \<lbrakk>?H1 x_1 (?H2 x_2); ?H3 x_3 x_2 = ?H4 x_1\<rbrakk> \<Longrightarrow> x_3 = ?H4 x_1 | lemma_command | ###symbols
Term.subst_apply_term
Term.term.Var
Set.not_member
Term.range_vars
Term.subst_range
Term.term.funs_term
More_Unification.subst_elim
Set.range
###defs
abbreviation subst_apply_term :: "('f, 'v) term \<Rightarrow> ('f, 'v, 'w) gsubst \<Rightarrow> ('f, 'w) term" (infixl "\<cdot>" 67)
where "subst_apply_term... | ###output
lemma subst_Var_notin_img: "x \<notin> range_vars s \<Longrightarrow> t \<cdot> s = Var x \<Longrightarrow> t = Var x"
###end |
Complex_Analysis/Contour_Integration | Contour_Integration.contour_integral_linepath_Reals_eq | lemma contour_integral_linepath_Reals_eq:
fixes a b :: complex and f :: "complex \<Rightarrow> complex"
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))" | ?a \<in> \<real> \<Longrightarrow> ?b \<in> \<real> \<Longrightarrow> Re ?a < Re ?b \<Longrightarrow> contour_integral (linepath ?a ?b) ?f = integral {Re ?a..Re ?b} (\<lambda>x. ?f (complex_of_real x)) | \<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1; ?H2 x_1 < ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) x_3 = ?H5 (?H6 (?H2 x_1) (?H2 x_2)) (\<lambda>y_0. x_3 (?H7 y_0)) | lemma_command | ###symbols
Complex.complex_of_real
Path_Connected.linepath
Contour_Integration.contour_integral
Henstock_Kurzweil_Integration.integral
Real_Vector_Spaces.Reals
Complex.complex.Re
Set_Interval.ord_class.atLeastAtMost
###defs
abbreviation complex_of_real :: "real \<Rightarrow> complex"
where "complex_of_real \<equiv> o... | ###output
lemma contour_integral_linepath_Reals_eq:
fixes a b :: complex and f :: "complex \<Rightarrow> complex"
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
###end |
CZH_Universal_Constructions/czh_ucategories/CZH_UCAT_Universal | CZH_UCAT_Universal.ntcf_ua_of_components(4) | lemma ntcf_ua_of_components:
shows "ntcf_ua_of \<alpha> \<FF> c r u\<lparr>NTMap\<rparr> = (\<lambda>d\<in>\<^sub>\<circ>\<FF>\<lparr>HomDom\<rparr>\<lparr>Obj\<rparr>. umap_of \<FF> c r u d)"
and "ntcf_ua_of \<alpha> \<FF> c r u\<lparr>NTDom\<rparr> = Hom\<^sub>O\<^sub>.\<^sub>C\<^bsub>\<alpha>\<^esub>\<FF>\<lpa... | ntcf_ua_of ?\<alpha> ?\<FF> ?c ?r ?u\<lparr>NTDGDom\<rparr> = ?\<FF>\<lparr>HomDom\<rparr> | ?H1 (?H2 x_1 x_2 x_3 x_4 x_5) ?H3 = ?H1 x_2 ?H4 | lemma_command | ###symbols
ZFC_Cardinals.VLambda
ZFC_Cardinals.app
CZH_DG_TDGHM.NTDGDom
CZH_UCAT_Universal.ntcf_ua_of
CZH_DG_TDGHM.NTMap
CZH_DG_DGHM.HomDom
###defs
definition VLambda :: "V \<Rightarrow> (V \<Rightarrow> V) \<Rightarrow> V"
where "VLambda A b \<equiv> set ((\<lambda>x. \<langle>x,b x\<rangle>) ` elts A)"
definition a... | ###output
lemma ntcf_ua_of_components:
shows "ntcf_ua_of \<alpha> \<FF> c r u\<lparr>NTMap\<rparr> = (\<lambda>d\<in>\<^sub>\<circ>\<FF>\<lparr>HomDom\<rparr>\<lparr>Obj\<rparr>. umap_of \<FF> c r u d)"
and "ntcf_ua_of \<alpha> \<FF> c r u\<lparr>NTDom\<rparr> = Hom\<^sub>O\<^sub>.\<^sub>C\<^bsub>\<alpha>\<^esub>... |
List-Infinite/CommonSet/SetIntervalCut | SetIntervalCut.cut_less_absorb | null | ?I \<down>< ?t \<down>< ?t = ?I \<down>< ?t | ?H1 (?H1 x_1 x_2) x_2 = ?H1 x_1 x_2 | lemma_command | ###symbols
SetIntervalCut.cut_less
Set.Collect
###defs
| ###output
None
###end |
Promela/Promela | PromelaDatastructures.comparator_variable_simps(2) | null | comparator_variable (Var ?x ?xa) (VArray ?yb ?yc ?yd) = Lt | ?H1 (?H2 x_1 x_2) (?H3 x_3 x_4 x_5) = ?H4 | lemma_command | ###symbols
PromelaAST.AST.unOp.case_unOp
Comparator.order.Lt
PromelaDatastructures.comparator_variable
Promela.pollCheck
PromelaAST.AST.recvArg.RecvArgVar
PromelaDatastructures.variable.Var
Promela.modProcArg
PromelaDatastructures.variable.VArray
###defs
datatype variable = Var varType integer
| VArra... | ###output
None
###end |
Combinatorics_Words/Submonoids | Submonoids.sing_lists_exp_len | lemma sing_lists_exp_len: "ws \<in> lists {x} \<Longrightarrow> [x]\<^sup>@\<^bold>|ws\<^bold>| = ws" | ?ws \<in> lists {?x} \<Longrightarrow> [?x] \<^sup>@ \<^bold>|?ws\<^bold>| = ?ws | x_1 \<in> ?H1 (?H2 x_2 ?H3) \<Longrightarrow> ?H4 (?H5 x_2 ?H6) (?H7 x_1) = x_1 | lemma_command | ###symbols
Set.insert
CoWBasic.list_power
Set.empty
List.lists
List.list.Nil
List.list.Cons
List.length
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
primrec list_power :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list" (in... | ###output
lemma sing_lists_exp_len: "ws \<in> lists {x} \<Longrightarrow> [x]\<^sup>@\<^bold>|ws\<^bold>| = ws"
###end |
JinjaDCI/BV/BVSpecTypeSafe | BVSpecTypeSafe.Calling_correct | lemma Calling_correct:
fixes \<sigma>' :: jvm_state
assumes wtprog: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
assumes mC: "P \<turnstile> C sees M,b:Ts\<rightarrow>T=(mxs,mxl\<^sub>0,ins,xt) in C"
assumes s': "Some \<sigma>' = exec (P, None, h, (stk,loc,C,M,pc,ics)#frs, sh)"
assumes cf: "P,\<Phi> \<turnstile> (No... | wf_jvm_prog\<^bsub>?\<Phi>\<^esub> ?P \<Longrightarrow> ?P \<turnstile> ?C sees ?M, ?b : ?Ts\<rightarrow>?T = (?mxs, ?mxl\<^sub>0, ?ins, ?xt) in ?C \<Longrightarrow> \<lfloor>?\<sigma>'\<rfloor> = exec (?P, None, ?h, (?stk, ?loc, ?C, ?M, ?pc, ?ics) # ?frs, ?sh) \<Longrightarrow> ?P,?\<Phi> |- (... | \<lbrakk>?H1 x_1 x_2; ?H2 x_2 x_3 x_4 x_5 x_6 x_7 (x_8, x_9, x_10, x_11) x_3; ?H3 x_12 = ?H4 (x_2, ?H5, x_13, ?H6 (x_14, x_15, x_3, x_4, x_16, x_17) x_18, x_19); ?H7 x_2 x_1 (?H5, x_13, ?H6 (x_14, x_15, x_3, x_4, x_16, x_17) x_18, x_19); ?H8 (?H9 x_2 x_13 x_14 x_15 x_3 x_4 x_16 x_17 x_18 x_19) = ?H5; x_17 = ?H10 ... | lemma_command | ###symbols
Option.option.Some
BVSpec.wf_jvm_prog_phi
Product_Type.prod.fst
JVMState.init_call_status.Calling
JVMExec.exec
List.list.Cons
JVMExec.exec_step
Option.option.None
BVConform.correct_state
TypeRel.Method
###defs
datatype 'a option =
None
| Some (the: 'a)
definition "prod = {f. \<exists>a b. f = Pair_Rep ... | ###output
lemma Calling_correct:
fixes \<sigma>' :: jvm_state
assumes wtprog: "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
assumes mC: "P \<turnstile> C sees M,b:Ts\<rightarrow>T=(mxs,mxl\<^sub>0,ins,xt) in C"
assumes s': "Some \<sigma>' = exec (P, None, h, (stk,loc,C,M,pc,ics)#frs, sh)"
assumes cf: "P,\<Phi> \<turn... |
Ordinary_Differential_Equations/Refinement/Refine_ScaleR2 | Refine_ScaleR2.scaleR2_empty | lemma scaleR2_empty[simp]: "scaleR2 l u {} = {}" | scaleR2 ?l ?u {} = {} | ?H1 x_1 x_2 ?H2 = ?H2 | lemma_command | ###symbols
Refine_ScaleR2.op_single_inter_ivl
Set.empty
Refine_ScaleR2.op_inter_fst
List.drop
Refine_ScaleR2.scaleR2
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
drop_Nil: "drop n [] = []" |
drop_Cons: "drop n (x # xs) = (... | ###output
lemma scaleR2_empty[simp]: "scaleR2 l u {} = {}"
###end |
Gauss_Jordan/Bases_Of_Fundamental_Subspaces_IArrays | Bases_Of_Fundamental_Subspaces_IArrays.vec_to_iarray_basis_col_space | null | vec_to_iarray ` basis_col_space ?A = basis_col_space_iarrays (matrix_to_iarray ?A) | ?H1 ?H2 (?H3 x_1) = ?H4 (?H5 x_1) | lemma_command | ###symbols
Set.image
Bases_Of_Fundamental_Subspaces_IArrays.basis_col_space_iarrays
Matrix_To_IArray.matrix_to_iarray
Gauss_Jordan_IArrays.rank_iarray
Matrix_To_IArray.vec_to_iarray
Bases_Of_Fundamental_Subspaces.basis_col_space
###defs
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b se... | ###output
None
###end |
JiveDataStoreModel/Isabelle_Store/Location | Location.isStaticLoc_simps(3) | lemma isStaticLoc_simps [simp]:
"isStaticLoc (objLoc cf a) = False"
"isStaticLoc (staticLoc f) = True"
"isStaticLoc (arrLenLoc T a) = False"
"isStaticLoc (arrLoc T a i) = False" | isStaticLoc (arrLenLoc ?T ?a) = False | ?H1 (?H2 x_1 x_2) = False | lemma_command | ###symbols
Location.isStaticLoc
Attributes.dtype
Location.Location.staticLoc
Location.Location.arrLenLoc
Location.ref
###defs
| ###output
lemma isStaticLoc_simps [simp]:
"isStaticLoc (objLoc cf a) = False"
"isStaticLoc (staticLoc f) = True"
"isStaticLoc (arrLenLoc T a) = False"
"isStaticLoc (arrLoc T a i) = False"
###end |
Dirichlet_Series/Dirichlet_Product | Dirichlet_Product.dirichlet_prod_assoc_aux1 | lemma dirichlet_prod_assoc_aux1:
assumes "n > 0"
shows "dirichlet_prod f (dirichlet_prod g h) n =
(\<Sum>(a, b, c)\<in>{(a, b, c). a * b * c = n}. f a * g b * h c)" | 0 < ?n \<Longrightarrow> dirichlet_prod ?f (dirichlet_prod ?g ?h) ?n = (\<Sum>(a, b, c)\<in>{(a, b, c). a * b * c = ?n}. ?f a * ?g b * ?h c) | ?H1 < x_1 \<Longrightarrow> ?H2 x_2 (?H2 x_3 x_4) x_1 = ?H3 (?H4 (\<lambda>y_0. ?H5 (\<lambda>y_1 y_2. ?H6 (?H6 (x_2 y_0) (x_3 y_1)) (x_4 y_2)))) (?H7 (?H8 (\<lambda>y_3. ?H9 (\<lambda>y_4 y_5. ?H10 (?H10 y_3 y_4) y_5 = x_1)))) | lemma_command | ###symbols
Product_Type.prod.case_prod
Dirichlet_Product.dirichlet_prod
Groups.times_class.times
Set.Collect
Groups_Big.comm_monoid_add_class.sum
Nat.Suc
Groups.one_class.one
Groups.zero_class.zero
Groups.uminus_class.uminus
###defs
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
definition dirichle... | ###output
lemma dirichlet_prod_assoc_aux1:
assumes "n > 0"
shows "dirichlet_prod f (dirichlet_prod g h) n =
(\<Sum>(a, b, c)\<in>{(a, b, c). a * b * c = n}. f a * g b * h c)"
###end |
Special_Function_Bounds/Log_CF_Bounds | Log_CF_Bounds.ln_upper_11_pos | lemma ln_upper_11_pos:
assumes "1 \<le> x" shows "ln(x) \<le> ln_upper_11 x" | 1 \<le> ?x \<Longrightarrow> ln ?x \<le> ln_upper_11 ?x | ?H1 \<le> x_1 \<Longrightarrow> ?H2 x_1 \<le> ?H3 x_1 | lemma_command | ###symbols
Transcendental.ln_class.ln
Log_CF_Bounds.ln_upper_11
Groups.one_class.one
###defs
class ln = real_normed_algebra_1 + banach +
fixes ln :: "'a \<Rightarrow> 'a"
assumes ln_one [simp]: "ln 1 = 0"
definition ln_upper_11 :: "real \<Rightarrow> real"
where "ln_upper_11 x \<equiv>
(5*x^5 + 647*x^4... | ###output
lemma ln_upper_11_pos:
assumes "1 \<le> x" shows "ln(x) \<le> ln_upper_11 x"
###end |
AODV/variants/a_norreqid/A_Aodv | A_Aodv_Data.vD_Some | null | ?dip \<in> vD ?rt \<Longrightarrow> \<exists>dsn dsk hops nhip pre. ?rt ?dip = Some (dsn, dsk, val, hops, nhip, pre) | x_1 \<in> ?H1 x_2 \<Longrightarrow> \<exists>y_0 y_1 y_2 y_3 y_4. x_2 x_1 = ?H2 (y_0, y_1, ?H3, y_2, y_3, y_4) | lemma_command | ###symbols
Aodv_Basic.val
Option.option.Some
A_Aodv_Data.vD
###defs
abbreviation val where "val \<equiv> Valid"
datatype 'a option =
None
| Some (the: 'a)
| ###output
None
###end |
Launchbury/C | Complete_Lattices.INF1_I | null | (\<And>x. x \<in> ?A \<Longrightarrow> ?B x ?b) \<Longrightarrow> Inf (?B ` ?A) ?b | (\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> x_2 y_0 x_3) \<Longrightarrow> ?H1 (?H2 x_2 x_1) x_3 | lemma_command | ###symbols
Set.image
Rat.field_char_0_class.of_rat
Complete_Lattices.Inf_class.Inf
###defs
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
class Inf =
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter> _" [900]... | ###output
None
###end |
ConcurrentIMP/CIMP_vcg | CIMP_vcg.prerun_reachable_state | lemma prerun_reachable_state:
assumes "prerun sys \<sigma>"
shows "reachable_state sys (\<sigma> i)" | prerun ?sys ?\<sigma> \<Longrightarrow> reachable_state ?sys (?\<sigma> ?i) | ?H1 x_1 x_2 \<Longrightarrow> ?H2 x_1 (x_2 x_3) | lemma_command | ###symbols
CIMP_lang.prerun
CIMP_vcg.reachable_state
###defs
definition
prerun :: "('answer, 'location, 'proc, 'question, 'state, 'ext) pre_system_ext
\<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) system_state seq_pred"
where
"prerun sys = ((\<lambda>\<sigma>. initial_state sys (GST (\<sigma... | ###output
lemma prerun_reachable_state:
assumes "prerun sys \<sigma>"
shows "reachable_state sys (\<sigma> i)"
###end |
Smooth_Manifolds/Projective_Space | Projective_Space.scaleR_scaleR_nonzero | lemma scaleR_scaleR_nonzero[simp]: "b \<noteq> 0 \<Longrightarrow> scaleR a (scaleR b x) = scaleR (a * b) (x::_ nonzero)" | ?b \<noteq> 0 \<Longrightarrow> ?a *\<^sub>R ?b *\<^sub>R ?x = (?a * ?b) *\<^sub>R ?x | x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 x_2 (?H2 x_1 x_3) = ?H2 (?H3 x_2 x_1) x_3 | lemma_command | ###symbols
Groups.zero_class.zero
Real_Vector_Spaces.scaleR_class.scaleR
Groups.times_class.times
Real_Vector_Spaces.norm_class.norm
Product_Type.prod.snd
Extended_Nat.infinity_class.infinity
###defs
class zero =
fixes zero :: 'a ("0")
class scaleR =
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr... | ###output
lemma scaleR_scaleR_nonzero[simp]: "b \<noteq> 0 \<Longrightarrow> scaleR a (scaleR b x) = scaleR (a * b) (x::_ nonzero)"
###end |
Prime_Number_Theorem/Prime_Counting_Functions | Prime_Counting_Functions.prod_primes_upto_less | lemma prod_primes_upto_less:
defines "F \<equiv> (\<lambda>n. (\<Prod>{p::nat. prime p \<and> p \<le> n}))"
shows "n > 0 \<Longrightarrow> F n < 4 ^ n" | 0 < ?n \<Longrightarrow> \<Prod>{p. prime p \<and> p \<le> ?n} < 4 ^ ?n | ?H1 < x_1 \<Longrightarrow> ?H2 (?H3 (\<lambda>y_0. ?H4 y_0 \<and> y_0 \<le> x_1)) < ?H5 (?H6 (?H7 (?H7 ?H8))) x_1 | lemma_command | ###symbols
Groups_Big.comm_monoid_mult_class.Prod
Num.num.Bit0
Groups.zero_class.zero
Num.numeral_class.numeral
Num.num.One
Factorial_Ring.normalization_semidom_class.prime
Power.power_class.power
Set.Collect
###defs
datatype num = One | Bit0 num | Bit1 num
class zero =
fixes zero :: 'a ("0")
primrec numeral :: "num... | ###output
lemma prod_primes_upto_less:
defines "F \<equiv> (\<lambda>n. (\<Prod>{p::nat. prime p \<and> p \<le> n}))"
shows "n > 0 \<Longrightarrow> F n < 4 ^ n"
###end |
Complete_Non_Orders/Well_Relations | Well_Relations.omega_chainI | lemma omega_chainI:
fixes f :: "nat \<Rightarrow> 'a"
assumes "monotone (\<le>) r f" "range f = A" shows "omega_chain A r" | monotone (\<le>) ?r ?f \<Longrightarrow> range ?f = ?A \<Longrightarrow> omega_chain ?A ?r | \<lbrakk>?H1 (\<le>) x_1 x_2; ?H2 x_2 = x_3\<rbrakk> \<Longrightarrow> ?H3 x_3 x_1 | lemma_command | ###symbols
Set.insert
Fun.monotone
Set.range
Binary_Relations.irreflexive
Well_Relations.omega_chain
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
abbreviation monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow... | ###output
lemma omega_chainI:
fixes f :: "nat \<Rightarrow> 'a"
assumes "monotone (\<le>) r f" "range f = A" shows "omega_chain A r"
###end |
Analysis/Equivalence_Measurable_On_Borel | Equivalence_Measurable_On_Borel.indicator_measurable_on | null | ?S \<in> sets lebesgue \<Longrightarrow> indicat_real ?S measurable_on UNIV | x_1 \<in> ?H1 ?H2 \<Longrightarrow> ?H3 (?H4 x_1) ?H5 | lemma_command | ###symbols
Set.UNIV
Sigma_Algebra.sets
Fields.inverse_class.inverse_divide
Henstock_Kurzweil_Integration.integrable_on
Indicator_Function.indicat_real
Equivalence_Lebesgue_Henstock_Integration.absolutely_integrable_on
Equivalence_Measurable_On_Borel.measurable_on
Product_Type.prod.fst
Lebesgue_Measure.lebesgue
###defs
... | ###output
None
###end |
Metalogic_ProofChecker/Term_Subst | Term_Subst.subst_typ'_simulates_tsubst_gen | lemma subst_typ'_simulates_tsubst_gen: "tsubst t \<rho>
= subst_typ' (map (\<lambda>(x,y).((x,y), \<rho> x y)) (SOME l . distinct l \<and> tvs t \<subseteq> set l)) t" | tsubst ?t ?\<rho> = subst_typ' (map (\<lambda>(x, y). ((x, y), ?\<rho> x y)) (SOME l. distinct l \<and> tvs ?t \<subseteq> set l)) ?t | ?H1 x_1 x_2 = ?H2 (?H3 (?H4 (\<lambda>y_0 y_1. ((y_0, y_1), x_2 y_0 y_1))) (?H5 (\<lambda>y_2. ?H6 y_2 \<and> ?H7 (?H8 x_1) (?H9 y_2)))) x_1 | lemma_command | ###symbols
Term.tvs
List.distinct
List.list.map
Core.tsubst
Product_Type.prod.case_prod
Core.typ.Tv
List.list.set
Term_Subst.subst_typ'
Set.subset_eq
Hilbert_Choice.Eps
Core.tsubstT
###defs
fun tvs :: "term \<Rightarrow> (variable \<times> sort) set" where
"tvs (Ct _ T) = tvsT T"
| "tvs (Fv _ T) = tvsT T"
| "tvs (Bv ... | ###output
lemma subst_typ'_simulates_tsubst_gen: "tsubst t \<rho>
= subst_typ' (map (\<lambda>(x,y).((x,y), \<rho> x y)) (SOME l . distinct l \<and> tvs t \<subseteq> set l)) t"
###end |
Combinatorial_Enumeration_Algorithms/n_Subsets | n_Subsets.n_subset_enum_correct_aux1 | lemma n_subset_enum_correct_aux1:
"\<lbrakk>distinct xs; length ys = length xs\<rbrakk>
\<Longrightarrow> set (filter_bool_list ys xs) \<in> n_subsets (set xs) (count_list ys True)" | distinct ?xs \<Longrightarrow> length ?ys = length ?xs \<Longrightarrow> set (filter_bool_list ?ys ?xs) \<in> n_subsets (set ?xs) (count_list ?ys True) | \<lbrakk>?H1 x_1; ?H2 x_2 = ?H3 x_1\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_2 x_1) \<in> ?H6 (?H4 x_1) (?H7 x_2 True) | lemma_command | ###symbols
List.count_list
n_Subsets.n_subsets
List.length
List.distinct
List.list.set
Filter_Bool_List.filter_bool_list
###defs
primrec count_list :: "'a list \<Rightarrow> 'a \<Rightarrow> nat" where
"count_list [] y = 0" |
"count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)"
definition n_su... | ###output
lemma n_subset_enum_correct_aux1:
"\<lbrakk>distinct xs; length ys = length xs\<rbrakk>
\<Longrightarrow> set (filter_bool_list ys xs) \<in> n_subsets (set xs) (count_list ys True)"
###end |
CryptHOL/GPV_Expectation | GPV_Expectation.ennreal_add_partial_function_mono | lemma ennreal_add_partial_function_mono [partial_function_mono]:
"\<lbrakk> monotone (fun_ord (\<le>)) (\<le>) f; monotone (fun_ord (\<le>)) (\<le>) g \<rbrakk>
\<Longrightarrow> monotone (fun_ord (\<le>)) (\<le>) (\<lambda>x. f x + g x :: ennreal)" | monotone (fun_ord (\<le>)) (\<le>) ?f \<Longrightarrow> monotone (fun_ord (\<le>)) (\<le>) ?g \<Longrightarrow> monotone (fun_ord (\<le>)) (\<le>) (\<lambda>x. ?f x + ?g x) | \<lbrakk>?H1 (?H2 (\<le>)) (\<le>) x_1; ?H1 (?H2 (\<le>)) (\<le>) x_2\<rbrakk> \<Longrightarrow> ?H1 (?H2 (\<le>)) (\<le>) (\<lambda>y_0. ?H3 (x_1 y_0) (x_2 y_0)) | lemma_command | ###symbols
Groups.plus_class.plus
Partial_Function.fun_ord
Generative_Probabilistic_Value.bind_gpv
Fun.monotone
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
definition "fun_ord ord f g \<longleftrightarrow> (\<forall>x. ord (f x) (g x))"
primcorec bind_gpv :: "('a, 'out, ... | ###output
lemma ennreal_add_partial_function_mono [partial_function_mono]:
"\<lbrakk> monotone (fun_ord (\<le>)) (\<le>) f; monotone (fun_ord (\<le>)) (\<le>) g \<rbrakk>
\<Longrightarrow> monotone (fun_ord (\<le>)) (\<le>) (\<lambda>x. f x + g x :: ennreal)"
###end |
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